Trying to install OS X Lion, machine hangs on boot

I have a Macbook Air A1237 that I'm trying to install Lion on, but when it restarts to complete installation, it just hangs on a white screen. I've used a usb stick that I put Lion on, I've downloaded Lion from the app store and installed it that way, but it always does the same thing. I can only install as high as 10.6.8 on the laptop. I've read that it can go as high as 10.7.5 and that's all I'm trying to do here.

I do not have a complete history on the laptop. I bought it broken with the intention to fix and sell. I replaced the screen and the lvds cable. I've been working on this thing for the better part of a year trying everything I know how to do to fix it. SMC reset, PRAM reset, replaced the hard drive with a brand new Super Talent 128gb ssd. Nothing has worked. Is there anything I'm missing?

My fear with this is that the logic board needs replaced.