I believe the light is replaceable, assuming you can find the part.I once researched this for my own SP, but I never actually ordered the parts.
You will need a tri-wing screwdriver. (Here's one on Amazon.)
You will also need the light, depending on which model you have. (Looking now, I'm having a hard time finding the replacement for the light.) Some GBA SPs had a front light (the original), and some of the newer ones had a backlight. Because there's so much coverage of the issue, searching online for "Gameboy Advance SP backlight" might not get you very far in terms of the part.
Here's a Google Search to get you started. If you get any more info on the parts, let me know in the comments.
I just walked into a local repair shop - which sells all kinds of used games, systems and such - I was told that the used Gameboy Advance SPs are around thirty bucks, so a repair isn't worth it. When I told him that I like the "thrill" of opening up my old gameboy, the guy said that if he gets one in really bad condition, he'd give me a good price on it, for the part. The point being, you should try the same. You might be able to get the part for cheap from a local game repair shop.