Sure you could upgrade your Core Solo to a Core 2 processor, but it would not be very cost effective.
Let me preface, if you still have AppleCare on this machine I do not recommend modifying your machine in any way, as processor upgrading voids warranty. If one was to upgrade, you'd have to use either a Socket M Yonah or Merom generation processor. The Mac Mini uses a pinned processor, unlike most traditional PC processors do today.
There are lots of Mac Mini hack and modification sites all over the net. But by the time you've purchased the processor, thermal paste, and tools required to do this modification, it would be less to purchase a current Mac Mini retail (and you'd have a fresh warranty too).
If its performance you're after, consider upgrading your memory. PC2-5300 SO-DIMMs are cheap right now, upgrading to a gig or 2 gigs would boost things for ya.