i have an eee pc and had exactly the same problem.
earthworm jim, oddworld and duke nukem manhattan project are perfect. in fact any platformer works great. plus those are both amazing games.
i mostly had trouble using the trackpad in games that required quick, accurate mouse movement, but i found that almost all turn-based strategy games and p-n-c adventures play fine--you have enough time to position the mouse and click where you want to.
i ended up getting a saitek p2500 gamepad for like $8. just go to amazon and look for a gamepad or optical mouse for $5. i have twenty gogs now that i play on my eee pc regularly!
props to my fellow netbook gamer(s?)!!

the old tomb raider games were played just with the keyboard. though i'd check that you can remap the keys, as they tended to rely on the arrow keys, which can be in weird places on laptops.
Doom, Doom2, Heretic, Hexen, Dark Forces are all totally playable with just the keyboard.

An awesome game that doesn't use a keyboard is Heart of Darkness, if that came here I'd buy it again even though I already own it for the PSX.

TrackMania. I have always played it with a keyboard, and it works perfectly.

*gasp*
The classic Tomb Raiders. Of course. I would sell my own body if it meant GOG would sell Tomb Raider.

Are you Hot and Female? I couldn't guarantee Tomb Raider on GoG though.
And I know. But someone had to say it.
Karl

JAZZ JACKRABBIT 2
GET IT NOW

The mentioned Trackmania and any 2.5D FPS game. A lot of the old old RPGs work too.
again: Pretty much every rougelike out there. Mouse support is fairly recent for these and not required at all. I spent a good part of the afternoon playing Z+angband with only a keyboard*, no sense doing work when the building is pretty much empty :)
* (as some...one stole my last working mouse it appears, more on order thankfully)

Resident Evil 1-2-3-4 don't need a mouse on PC....

In Cold Blood is another good no-mouse game.
Some of the EA sports games don't need a mouse, the older ones would probably be playable on a netbook
Any action game where you don't have or need control over the camera would probably work, I can almost imagine playing beyond good & evil without a mouse
There's also the really old games like the Ultima's & text adventures. C64 & Amiga emulation would probably suit you very well for a netbook, its one of the things I plan to do when I can afford a big SD card for mine

I hope you're not picky because I'm neither. Is that a Tomb Raider II CD in your pocket or are you just... carrying a CD in your pocket?
Cheers everyone for the suggestions! I'm going to figure out a way to get my Tomb Raider III CD onto my netbook and I'll give a crack at the emulator business.
I'll see if there are demos for Defense Grid to give that a go too. I'm a sucker for tower defense.

If you want something more recent Braid is played with just the keyboard.

To answer my question and anyone's question for the next few years of my life download Spelunky.
I only recently found out about it but it's the most incredible experience evar.
Go get it
It's a platform adventure game with RPG elements and it's roguelike in the sense that it's completely insane and every level is randomly generated.

Ultima IV to VI are all perfectly good with no mouse.
Prince of Persia (the original)
Alone in the Dark (the original three)
Elite Plus
Flashback (I think!)
Old Sierra games (i.e. early Quest series, text input.)