I use Skin-So-Soft w/ SPF 30 from Avon on my son. He's 2 and will get bug bites anywhere we go if he doesn't have this stuff on. (I saw an Avon rep has already posted to you, so you can see what they have you might like.) My son knows the bugs bite him all the time, so he gets REALLY upset if he even sees a bug in the house. It can be a little aphid, accidentally brought in on a fresh cut rose, and you'd swear someone had a knife to my throat with the terror in his cry.
My doc said Deet is okay for short periods of time, and does work better than deet free, but shouldn't be used on a regular basis, for ANYONE. We went camping last summer for 17 days, and doc said it was fine for the trip, because he knows how sensitive my son is to bug bites. He just said not to put it on his hands, or any where with a direct opening into the body; like mouth, nose, ears, or buttocks. Doc said it was okay when my son was only 18 months. If it is just for short lengths of time, Deet is okay, but I still recommend Skin-So-Soft, especially those with SPF. (Wouldn’t you use sunscreen while he’s playing outside anyway?)
You can also put it on you, and just on your kid's cloths. If you're playing close to him, it may be enough to keep the bugs at bay, and none of it is actually on his skin.
Hope it helped. -J