At that cost, I'd either part it out or find a working logic board on eBay, but in general, I'm pretty hesitant to spend money on "known to fail" hardware. You'll kick the can for a bit, but that's about it.
What are you using the computer for? If it's pretty much entirely web, a couple hundred dollar Chromebook is a really good option.
If it's more general stuff, a Chromebook isn't a great option.
And, RWD, depending on the use, Linux isn't an option. I use it plenty, but it's not going to run things like Lightroom very well.