I'm doing that just now, just waiting on a replacement case.
You can get a good condition GBA for about £20 and the AGS-101 backlit screen + lcd cable for around £35-£40.
You just need to get the right cable type for your console, 32 or 40 pin. The screen sellers often show how to tell them apart.
I waited until my GBA arrived and then ordered the right screen.
You can also get complete replacement case/buttons/glass screen protector kits for around £15 in various colours.
I've opted for a dark clear case with SNES colour buttons, it looks cool in my head anyway.
A pre-modded GBA is around £70, although it looks fun to do yourself.
Once it's complete and working I'll probably try a lithium battery mod, using 2x GBA SP lithium batteries in parallel with a charging board so I can charge the GBA using a micro USB cable.