Sounds like an overheat issue. Sometime foreign containments and moisture can damage the logic board over time. Considering it spends most of its time in a pocket or next to a face or in your hand it is prone to the elements.
Try removing the logic board and seeing if anything is on the board (dust, calcium deposits from liquid damage, or anything out of the normal).
If you see anything get a disposable toothbrush with medium soft bristles and clean the board off. If this doesn't work consider replacing the board or upgrading to a 3g or 3gs from ebay.
Replacement iPhone 2G board around $40-$80
iPhone 3G around $160-$250
iPhone 3GS around $285-$400
*These prices are without contracts.
I recommend upgrading to at least a 3G since it will become increasingly difficult to find good parts for a 2G model. The 3GS is a fantastic phone, but if you are going to upgrade to that it will be expensive and make sense to consider the iPhone 4. I myself upgraded to the iPhone 4 with a two year contract for $199. Well worth the money spent.