Why does my iPhone 3G keep turning off by itself even when it's not dead?

My dad had his iPhone 3G and gave it to me when he had an upgrade. He never had the issue of it turning off by itself, but when he gave it to me, all of a sudden it decided to turn off by itself even when it was on like 80% still.

I took it in to the Apple Store near me and they said that I had too much "stuff" on it, mostly music.

So, I completely redid my phone. I reset it completely so that it looked like a completely new iPhone. And then downloaded only 12 apps and 195 songs. And, only about 50 pictures. (This is an 8 gig by the way)

Right now, there's less than 4 gigs on there so far meaning I still have more than half of my phone to fill. They guy at the store said it needs 2 gigs to run.

SO, it should have started working since theres way less stuff on there now, and more than enough space to run. Why does it keep turning off still though?

It's so annoying and inconvenient. If anyone could help it would be SO appreciated.

THANKS!!