[TouchPad Q&A, Help & Troubleshooting] ve6ay:
The stuff that's installed on there will lower the free space. The actual partition size is changed by installing any form of Android / Linux / anything except the actual WebOS.

What explorer are you using that shows it's only 7 GB in size ? Root Explorer shows that my SD Card has 3.75 used, 6.88 free (so 10.63 total). I have the 16GB Touchpad with just yesterday's official nightly installed. I connected my tablet while booted into WebOS, and by using lvm.static lvdisplay (and saving the result to a text file) I get the following:

--- Logical volume ---
LV Name /dev/store/root
VG Name store
LV UUID sila5d-dULD-b5A2-vaei-gVJL-HtoQ-wWgupV
LV Write Access read/write
LV Status available
# open 1
LV Size 568.00 MB
Current LE 71
Segments 1
Allocation inherit
Read ahead sectors auto
- currently set to 256
Persistent major 254
Persistent minor 0
Block device 254:0

--- Logical volume ---
LV Name /dev/store/var
VG Name store
LV UUID jy0sqK-iGmC-VzSM-3v1e-FsiT-MU6j-2LXF2A
LV Write Access read/write
LV Status available
# open 1
LV Size 64.00 MB
Current LE 8
Segments 1
Allocation inherit
Read ahead sectors auto
- currently set to 256
Persistent major 254
Persistent minor 1
Block device 254:1

--- Logical volume ---
LV Name /dev/store/update
VG Name store
LV UUID O0w4f2-4WTg-nQOk-Gzrx-3ZCt-HJsn-pzkWnl
LV Write Access read/write
LV Status available
# open 0
LV Size 16.00 MB
Current LE 2
Segments 1
Allocation inherit
Read ahead sectors auto
- currently set to 256
Persistent major 254
Persistent minor 2
Block device 254:2

--- Logical volume ---
LV Name /dev/store/log
VG Name store
LV UUID 0bAx4I-B16G-5h0T-ZuN1-0tXv-hOXz-2QVGY7
LV Write Access read/write
LV Status available
# open 1
LV Size 24.00 MB
Current LE 3
Segments 1
Allocation inherit
Read ahead sectors auto
- currently set to 256
Persistent major 254
Persistent minor 3
Block device 254:3

--- Logical volume ---
LV Name /dev/store/mojodb
VG Name store
LV UUID TRDKpT-jhpg-WiP5-jTmu-CqP8-LuHf-V79EZj
LV Write Access read/write
LV Status available
# open 1
LV Size 256.00 MB
Current LE 32
Segments 1
Allocation inherit
Read ahead sectors auto
- currently set to 256
Persistent major 254
Persistent minor 4
Block device 254:4

--- Logical volume ---
LV Name /dev/store/filecache
VG Name store
LV UUID hJRmQn-tynn-0wYy-A0pn-27En-iOLF-V3xwhd
LV Write Access read/write
LV Status available
# open 1
LV Size 136.00 MB
Current LE 17
Segments 1
Allocation inherit
Read ahead sectors auto
- currently set to 256
Persistent major 254
Persistent minor 5
Block device 254:5

--- Logical volume ---
LV Name /dev/store/media
VG Name store
LV UUID qF4wkF-DYew-lsgu-nZw2-X42W-mdRS-Dp3jOW
LV Write Access read/write
LV Status available
# open 1
LV Size 10.64 GB
Current LE 1362
Segments 1
Allocation inherit
Read ahead sectors auto
- currently set to 256
Persistent major 254
Persistent minor 6
Block device 254:6

--- Logical volume ---
LV Name /dev/store/swap
VG Name store
LV UUID bm5Tzx-sIg1-DHay-FDqg-so0o-JIl3-F0WMr4
LV Write Access read/write
LV Status available
# open 1
LV Size 512.00 MB
Current LE 64
Segments 1
Allocation inherit
Read ahead sectors auto
- currently set to 256
Persistent major 254
Persistent minor 7
Block device 254:7

--- Logical volume ---
LV Name /dev/store/cm-system
VG Name store
LV UUID Udkiug-8e79-r1ny-3KOC-v9os-HR14-T5trzY
LV Write Access read/write
LV Status available
# open 0
LV Size 304.00 MB
Current LE 38
Segments 1
Allocation inherit
Read ahead sectors auto
- currently set to 256
Block device 254:8

--- Logical volume ---
LV Name /dev/store/cm-cache
VG Name store
LV UUID LSKR0O-ViBX-kAjw-HOVx-5eO1-DYqf-bJ1T0t
LV Write Access read/write
LV Status available
# open 0
LV Size 200.00 MB
Current LE 25
Segments 1
Allocation inherit
Read ahead sectors auto
- currently set to 256
Block device 254:9

--- Logical volume ---
LV Name /dev/store/cm-data
VG Name store
LV UUID LXwDzu-ouay-ksUq-baho-57pu-c4AW-9AJbx6
LV Write Access read/write
LV Status available
# open 0
LV Size 1.50 GB
Current LE 192
Segments 1
Allocation inherit
Read ahead sectors auto
- currently set to 256
Block device 254:10

So, boot up, run Novaterm, and issue the lvm.static command and read what it shows you for your /media/card partition. And check for anything that isn't in the list I just posted.