You can definitely use 8GB of RAM in that computer.
IMO, for most people it's not really as beneficial in that machine to upgrade to 16 GB as it is for other machines. Its 2.4 GHz Core 2 Duo is fast enough for very basic surfing and office apps, and can definitely make use of 8 GB RAM, but once you go beyond that type of usage it really lags, so it likely will become much harder to take full advantage of the RAM, aside from app caching, unless you have very specific development requirements or something like that.
So for the price, with that machine, I'd go with 8 GB. That said, if you can find 16 GB cheap, it can't hurt to go with 16.
BTW, if you have light usage, 4 GB is actually usually OK. In my experience for that class of machine in modern times:
2 GB with hard drive: Completely unusable.
4 GB with SSD: Very usable, for light to light-moderate usage
8 GB with SSD: Very usable, for heavier usage
That is to say, if you don't already have an SSD, get that first and see how it goes. If you still need a boost, then go to 8 GB.