Realistically, your chances will be pretty low. That's because GPUs from this era (early-mid 2000s) are very prone to BGA solder failure; because of the many heat cycles during its lifetime, the solder balls under the GPU die or the memory chips will crack and you will get screen artifacts or nothing at all.
One temporary fix for this issue is the ubiquitous "stick'em in the oven" approach, which will reflow all the solder joints. But, as you know, this will not last for very long, maybe a few weeks up to a few months until the card dies again. I used to do this oftenly for my 8800GTS cards.
The proper way would be to get one of those BGA stencils on eBay and a bag of new solder balls for GPUs. You will then have to remove the chip and clean the area out with some desoldering wick and install the new solder balls. As you would imagine, this is a very complicated task, usually done by very expensive machines/tools (such as one from Louis Rossmann's shop).
Knowing all of this, I would still buy them for $1, just because you never know what you're going to get, you could end up with easier fixes, like bulged caps or blown fuses if you're lucky.