There are some rock-cut tombs at the incorrectly named "Tombs of the Kings", just north of Paphos that date to around 400BCE. The site is extensive and impressive and is great fun for kids to explore (even big kids). Allocate a few hours to explore it. You can get there (and back) by bus from Kato Paphos for 50p each way.
There are some good restaurants if you walk south for 10 minutes (or take the bus one stop for another 50p) , e.g. Fat Mama's (arrive early or you'll have to queue) or 5 mins to the north is the 5 star Venus Beach Hotel that is welcoming to non-residents; use any of their facilities or attend their evening shows at no charge. Check with Tassos on reception for a list of their special events. We went for their flambe evening and had a delicious flambed tiger prawn Thai-style curry. The hotel was due to undergo refurbishment over the winter, so it's possible that it may not be open again yet.
Alternatively, if you just want a lunch, head back to Kato Paphos and look for a restaurant on the harbour that has fresh grilled sardines on the blackboard.
Anyway, back to archaeology, I found the following that includes some references to neolithic sites:
http://members.aol.com/EastMedit/aja/aja98.html
