iPhone logo on and off loop; tried solutions but :(

So, I have my mothers iPhone 3gs which is still in her possession (uses it as an alarm clock). A while back, she told me it would stay on the apple logo for minutes until it decided to knock itself out. I replaced the battery since it was a few years old and I assumed that was the problem. But, now I am stuck with a apple logo that flashes on for one sec and then I am given a black screen for about five. I tried plugging the phone into iTunes, but no connection is made... however, it does charge from my computer. I've tried DFU mode as well, but no luck.

Does anyone have any solutions of suggestions?

Thanks,

Jon