Just buy a Radeon 9000 (non-Pro) or 7500. I know the 9000 is fanless, but check to see if the 7500 is, as well--if you're not gaming, there's no point in adding another noise source to your system. A cheap GF FX 5200 or GF4MX should also do well. I'd buy the cheapest of the four, but I'd get a 9000 over a 7500 and a 5200 over a GF4MX if the former are only a few dollars more than the latter--greater resale value. If you don't want to hink about reselling, then just go for the cheapest--look into a Radeon 7000, as well, as others have said.
A DVI adapter is something that attaches to a card's DVI-I connector and adapts it to a DB15 one, allowing you to plug in another CRT. Only DVI-I connectors allow for the adapter; DVI-D ones don't. You can tell the two apart because the -I has a cross with four holes on one end, while the -D has only a line. Take a look at some card pics at NewEgg to see.
Matrox is renowned for excellent signal quality with DB15 connectors (for CRT's) due to their premium digital-to-analog circuitry, but DVI adapters really level the field in terms of quality as the DAC circuitry is bypassed.
Basically any video card with a DVI connector should drive a 1280x1024 LCD; above that, you'll have to check to see if the card supports reduced blanking intervals.