Water World water park: The coolest water park ever, with dozens of slides, pools and other water-based attractions. I went their with my mates twice and spent a whole day there; we had a great time. It's definitely a refreshing change compared to the lazy beach time you get on Ayia Napa main beaches.
The water park is 7 minute drive from the main Ayia Napa street. Shouldn't cost you more than 6 CYP by taxi. Split it among 3-4 friends and it's a cheap price.
The only probelm I have with this place is the outrageous food price. So if you're planning a whoel day's stay (which I think you should!), try to grab a bite before you go, or at least take some snacks with you. Being a hungry fellow, it can take up to 15 CYP to fill my stomach! There are definitely cheaper alternatives!

Cavo Greco a beautiful cliff on the outskirts of Ayia Napa, to the south west, I think. Also, very close – perhaps 5 minutes by taxi. It's a smashing combination of cliffs and caves, blue water and huge rocks. Lots of people jump off the cliff, plunging into the water. At first it looks extra spooky, but once you've learned the habit, it's easier. A MUST SEE place if you want to do something different on your vacation which is just a few minutes away.

Bungee! Don't forget the Bungee jumps. Last time I was there, there was one bungee post but the lads there told me that another one should be opened soon.
Ask any local and they'll tell you exactly how to find it. It's within a walking distance from the main streets or beaches.

The bungee place we went to was a source of attraction for many people that just hang out there, drinking a cold beer and watching the horror on people's faces as they jump :).