IPod Touch 1st Gen charging rebooting loop

Try to completely disable Push Notifications in the Settings application. This might lead to the iPod not trying to maintain a WiFi connection all the time and thus increase battery life.

Did you try charging using another USB cable (the one for your iPhone, perhaps) and see if that made a difference? I’d recommend testing this first before replacing attempting to replace any components yourself.

That said, it’s most likely a battery issue.

Here’s a guide on how to replace the first-generation iPad’s battery yourself.