Question: I just bought a monitor for a relative's computer (Intel 82915G chipset) which should produce 1600x900. Though the current card can produce 1600x1200, there's no way (that I know of) to force it to produce 1600x900. 

Assuming buying a card would be a solution, how would one find a cheap video card that can do 1600x900? They don't seem to list the modes in most ads.

If I'm missing something in Windows, please let me know, but I've looked around quite a bit and tried many things (including unsupported resolutions, etc).

Answer: 
Hooray! I have found a cure for this problem.
I have an optiplex gx280 with the same chipset, Intel(R) 82915G/GV/910GL.
Windows XP pro
I also have an Acer X203H 1600x900 monitor that was not able to achieve this resolution.

You can solve this dilemma by using an awesome software called Powerstrip. Can be found here <http://entechtaiwan.com/util/ps.shtm>

Follow the instructions here
<http://forums.entechtaiwan.com/index.php?topic=24.0>

Hooray! Resolution desired has been acquired.

HINT:I hooked up this monitor to a PC that worked with it and took a screen shot of the working monitor stats to enter into the non-working pc.
Most of the newer video cards support all standard monitor resolutions. NVIDIA and ATI brand cards come with software suites that allow you to specify resolutions if they are not available via Windows. Newegg.com sells all kinds of video cards and you should be able to find a good card for pretty cheap there.