Q: iTunes error 29 due iPhone 3Gs 16gb restore

http://support.apple.com/kb/TS3694#error29

That error can be caused by security software (antivirus or firewall) on your computer interfering with the update. However, if the error occurred in an Apple store when they tried to restore it's a hardware problem.

Then you need to try with the antivirus and firewall off. If you still have a problem try another computer, preferably a Mac.

If reputable repair shop confirms Hardware Error then there are 2 possibilities, 1) battery incompatibility, need compatible battery, 2) other hardware error of charging circuitry requiring new iPhone. Steps to trouble shoot this iPhone, 1) follow Lawrence's recommendations, if fixed great, if still problem, 2) compatible battery replacement, if fixed great, if still problem, 3) get new phone.