<PAGE>   1
                                                                 EXHIBIT 10.43.6

THIS WARRANT AND THE SECURITIES ISSUABLE UPON THE EXERCISE HEREOF HAVE NOT BEEN
REGISTERED UNDER THE SECURITIES ACT OF 1933. THEY MAY NOT BE SOLD, OFFERED FOR
SALE, PLEDGED, HYPOTHECATED, OR OTHERWISE TRANSFERRED EXCEPT PURSUANT TO AN
EFFECTIVE REGISTRATION STATEMENT UNDER THE SECURITIES ACT OF 1933, OR AN OPINION
OF COUNSEL SATISFACTORY TO THE COMPANY THAT REGISTRATION IS NOT REQUIRED UNDER
SUCH ACT OR UNLESS SOLD PURSUANT TO RULE 144 UNDER SUCH ACT.

                        WARRANT TO PURCHASE COMMON STOCK

                                       OF

                               WEBSIDESTORY, INC.

                            VOID AFTER JUNE 30, 2003

This Warrant is issued to TA/Advent VIII, L.P. ("Holder") by WebSideStory, Inc.,
a California corporation ( the "Company"), on June 30, 2000. This Warrant is
issued pursuant to the terms of that certain Stock Purchase Agreement dated as
of the date hereof (the "Purchase Agreement") in connection with the Company's
issuance and sale of 1,754,034 shares of its Convertible Redeemable
Participating Preferred Stock and 11.66667 shares of its Redeemable Preferred
Stock.

1    Purchase of Shares. Subject to the terms and conditions hereinafter set
forth and compliance with applicable laws, the Holder is entitled, upon
surrender of this Warrant at the principal office of the Company (or at such
other place as the Company shall notify the Holder in writing), to purchase from
the Company for up to an aggregate purchase price of $713,125.37509 a number of
fully paid and nonassessable shares of Common Stock of the Company, (the
"Shares") at a price per share of $0.3538676 (the "Exercise Price"); provided,
however, that if the Conversion Price per share of the Convertible Redeemable
Participating Preferred Stock, as defined in the Company's Amended and Restated
Articles of Incorporation, as amended from time to time (the "Restated
Articles") is adjusted as provided in the Restated Articles, then the Exercise
Price shall equal the Conversion Price as adjusted.

2    Exercise Period. This Warrant shall be exercisable, in whole or in part
during the term commencing on April 1, 2001 (the "Warrant Exercisable Date") and
ending at 5:00 p.m. on June 1, 2003; provided, however, that if any of the
following events occurs before the Warrant Exercisable Date, then this Warrant
shall not be exercisable and shall become null and void: (a) a Qualified Public
Offering as defined in the Restated Articles; (b) the closing of the sale or
transfer of all or substantially all of the assets of the Company to another
entity (the "Acquiring Person") at a purchase price of $150,000,000.00 or more;
or (c) the closing of the acquisition of the Company at a purchase price of
$150,000,000.00 or more by an Acquiring Person by means of merger, consolidation
or other transaction or series of related transactions, resulting in the
exchange of the outstanding shares of the Company's capital stock such that the
stockholders of the Company prior to such transaction own, directly or
indirectly, less than 50% of the voting power of the surviving entity, this
Warrant shall, on the date of such closing, no longer be


                                       1
<PAGE>   2

exercisable and become null and void. Furthermore, if either of the following
events occurs after the Warrant Exercisable Date, then this Warrant shall no
longer be exercisable and shall become null and void: (i) the closing of the
sale or transfer of all or substantially all of the assets of the Company to an
Acquiring Person; or (ii) the closing of the acquisition of the Company by an
Acquiring Person by means of merger, consolidation or other transaction or
series of related transactions, resulting in the exchange of the outstanding
shares of the Company's capital stock such that the stockholders of the Company
prior to such transaction own, directly or indirectly, less than 50% of the
voting power of the surviving entity, this Warrant shall, on the date of such
closing, no longer be exercisable and become null and void. In the event of a
proposed transaction of the kind described in the preceding sentence, the
Company shall notify the Holder at least fifteen (15) days prior to the closing
of such event or transaction. Each of the transactions specified in clauses (i)
and (ii) of the preceding sentence is hereinafter referred to as a "Corporate
Transaction."

3    Method of Exercise. While this Warrant remains outstanding and exercisable
in accordance with Section 2 above, the Holder may exercise, in whole or in
part, the purchase rights evidenced hereby. Such exercise shall be effected by:

     (a)  the surrender of the Warrant, together with a duly executed copy of
          the form of Notice of Election attached hereto, to the Secretary of
          the Company at its principal offices; and

     (b)  the payment to the Company of an amount equal to the aggregate
          Exercise Price for the number of Shares being purchased.

4    Conversion. If the Company's Common Stock is listed for trading on a
national exchange, in lieu of exercising this Warrant pursuant to Section 3, the
Holder may elect to convert this Warrant, without the payment by the Holder of
any additional consideration, into shares of Common Stock equal to the value of
this Warrant (or the portion hereof being cancelled) by surrender of this
Warrant to the Secretary of the Company at its principal office together with
notice of such election, in which event the Company shall issue to the Holder a
number of shares of Common Stock computed using the following formula:

               Y(A-B)
          X=   ------
                  A

Where: X = The number of shares of Common Stock to be issued to the Holder
pursuant to this conversion.

Y = The number of Shares in respect of which the election to convert is made;

A = The fair market value of one share of the Common Stock at the time the
election to convert is made;

B = The exercise Price (as adjusted to the date of the conversion).

For purposes of this Section 4, the fair market value of one share of Common
Stock as of a particular day shall be determined by reference to the average bid
and ask price for Common


                                       2
<PAGE>   3

Stock at the close of trading over the five (5) immediately preceding days on
which the relevant market is open for trading.

5    Certificates for Shares. Upon the exercise of the purchase or conversion
rights evidenced by this Warrant, one or more certificates for the number of
Shares so purchased shall be issued as soon as practicable thereafter (with
appropriate restrictive legends, if applicable), and in any event within thirty
(30) days of the delivery of the subscription notice.

6    Issuance of Shares. The Company covenants that the Shares, when issued
pursuant to the exercise of this Warrant, will be duly and validly issued, fully
paid and nonassessable and free from all taxes, liens, and charges with respect
to the issuance thereof other than those created by or imposed upon Holder
thereof through no action by the Company.

7    No Fractional Shares or Scrip. No fractional shares or scrip representing
fractional shares shall be issued upon the exercise or conversion of this
Warrant, but in lieu of such fractional shares the Company shall make a cash
payment therefore on the basis of the Exercise Price then in effect.

8    No Stockholder Rights. Prior to exercise or conversion of this Warrant, the
Holder shall not be entitled to any rights of a stockholder with respect to the
Shares, including (without limitation) the right to vote such Shares, receive
dividends or other distributions thereon, exercise preemptive rights or to be
notified of stockholder meetings, and the Holder shall not be entitled to any
notice or other communication concerning the business or affairs of the Company.
However, nothing in this Section 9 shall limit the right of the Holder to be
provided the notices required under this Warrant.

9    Transfer of Warrant. Subject to compliance with applicable federal and
state securities laws any other contractual restrictions between the Company and
the Holder, this Warrant and all rights hereunder are transferable in whole or
in part by the Holder to any person or entity upon written notice to the
Company. The transfer shall be recorded on the books of the Company upon the
surrender of this Warrant, properly endorsed, to the company and its principal
offices, and the payment to the Company of all transfer taxes and other
government charges imposed on such transfer. In the event of a partial transfer,
the Company shall issue to the Holder one or more appropriate new warrants.

10   Successors and Assigns. The terms and provisions of this Warrant shall
inure to the benefit of, and be binding upon, the Company and the Holder and
their respective successors and assigns.

11   Amendments and Waivers. Any term of this Warrant may be amended and the
observance of any term of this Warrant may be waived (either generally or in a
particular instance and either retroactively or prospectively), with the written
consent of the Company and the holders of a majority of shares of Common Stock
issued or issuable upon exercise of Warrants issued pursuant to the Purchase
Agreement. Any waiver or amendment effected in accordance with this Section 12
shall be binding upon each holder of any Shares purchased under this Warrant at
the time outstanding, each future holder of all such Shares, and the Company.


                                       3
<PAGE>   4

12   Effect of Amendment or Waiver. The Holder acknowledges that by the
operation of Section 12 hereof, the holders of a majority of shares of Common
Stock issued or issuable upon exercise of Warrants issued pursuant to the
Purchase Agreement will have the right and power to diminish or eliminate all
rights of such Holder under this Warrant.

13   Notices. All notices required under this Warrant shall be deemed to have
been given or made for all purposes (i) upon personal delivery, (ii) upon
confirmation receipt that the communication was successfully sent to the
applicable number if sent by facsimile; (iii) one day after being sent, when
sent by professional overnight courier service, or (iv) five days after posting
when sent by registered or certified mail. Notices to the Company shall be sent
to the principal office of the Company (or at such other place as the Company
shall notify the Holder hereof in writing). Notices to the Holder shall be sent
to the address of the Holder on the books of the Company (or at such other place
as the Holder shall notify the Company hereof in writing).

14   Attorneys' Fees. If any action of law or equity is necessary to enforce or
interpret the terms of this Warrant, the prevailing party shall be entitled to
its reasonable attorneys' fees, costs and disbursements in addition to any other
relief to which it may be entitled.

15   Captions. The section and subsection headings of this Warrant are inserted
for convenience only and shall not constitute a part of this Warrant in
construing or interpreting any provision hereof.

16   Governing Law. This Warrant shall be governed by the laws of the State of
California as applied to agreements among California residents made and to be
performed entirely within the State of California.



                                   WEBSIDESTORY, INC.


                                   BY:   /s/ JOHN J. HENTRICH
                                      -----------------------------
                                       John J. Hentrich
                                       President and Chief Executive Officer


                                       4
<PAGE>   5

                               NOTICE OF EXERCISE

To:  WebSideStory

The undersigned hereby elects to (check applicable subsection):

____      (a) Purchase _____________ shares of Common Stock of WebSideStory,
          Inc. pursuant to the terms of the attached Warrant and payment of the
          Exercise Price required under such Warrant accompanies this notice;

OR

____      (b) Exercise the attached Warrant for (all of the shares) [________ of
          the shares] (cross out inapplicable phrase) purchasable under the
          Warrant pursuant to the conversion provisions of Section 4 of such
          Warrant.

The undersigned hereby represents and warrants that the undersigned is acquiring
such shares for its own account for investment purposes only, and not for resale
or with a view to distribution of such shares or any part thereof.

                                 WARRANTHOLDER:

                                 ---------------------------

                             By:
                                 ---------------------------
                           Name:
                                 ---------------------------
                          Title:
                                 ---------------------------

                    Address:
                            --------------------------------

                            --------------------------------

Date:
     -----------------
Name in which shares should be registered:

---------------------------------


                                       5