Don't go into the city. First, it will cost you about $16 round trip on BART, since they've decided that they can rip off airport passengers. Second, the trip to the city will just be stressful on very noisy and dirty trains. But most importantly, you won't have enough time to really enjoy yourself - maybe 2 hours? That's not quite enough.
Have you been to In-n-Out (local West Coast burger chain) before? If not, that could be a fun little trip. It's quite close to the airport, and you can either take the Airtrain and walk a little bit to get to it or you can take a taxi (probably for around $16 round trip, too!).
Then you just have another hour to kill at the airport, which isn't bad - especially since you will be in HI soon enough!