Q:

Safecracker Return Message on Form Submission

Suppose I have a Safecracker form that will edit channel entries.  Once the entry is edited I want to display something like "Thanks for editing your entry" on the edit entry page.  How would I go about displaying a thank you message on successful form submission?  I do not want to be redirected to another page. 
Here is example code. 
{exp:safecracker channel="channel_name" return="channel_name/edit/ENTRY_ID" entry_id="{segment_3}"}

Title:<input name="title" type="text" value="{title}">
<br/><input type="submit" value="Submit">

{/exp:safecracker}

A:

It sounds like you are looking for an AJAX-style contact form where you display the success or failure message on the same page without a refresh.  I might suggest this article: Creating a modern AJAX contact form in ExpressionEngine.  It also mentions the difference between adding AJAX to EE's native contact form and adding AJAX to other form add-ons like Solspace's Freeform. 
Alternatively, you can reload the same page/template but add another URL segment, like:
return="channel_name/edit/ENTRY_ID/success"

Then you can display a success message on that same template when the segment matches:
{if segment_4 == "success"}
    <p>Success message here</p>
{/if}

This does not offer inline validation, like AJAX will, but it is a simple option.