The most likely problem is bad capacitors on the circuit board. Sega put cheapies in to keep costs down, and they'd fail fairly quickly as a result.
If you're good with soldering, you can replace the capacitors yourself. EBay has plenty of sellers who offer replacement sets, some with illustrated instructions.
http://www.ebay.com/sch/i...mp;_sacat=0&_from=R40That's the route I went to fix my two Game Gears, although I eventually paid someone to put 'em in for me.