My son, 8, gets the same thing. Its not the bites that are that big, its the swelling allergic reaction. One of my daughters has a bug bite problem, she will get bit many times, but not have the swelling reaction.It is strange to see them get that big and there is really not much you can do to keep them from getting like that.
However, there are some things you can do to avoid them getting bit. I have tried the sprays and rubs, they work a little, but not much. I am currently putting out citronella candles on my porch and in our yard to chase away about 1/2 of our bugs, but not enough.
A couple years ago I found bug bracelets at Walmart, BugOff! (deet free), in the seasonal area, which right now, they have grills, plants, garden supplies, and bug sprays. I stock up on them whenever I can, with 4 kids, they work better than anything else I've ever tried.
If your son is getting bit while he sleeps, I would say his room, or wherever he is sleeping when he gets bit, needs a real good cleaning! And don't forget to spray for bugs in any cracks and under beds. I live in an old house with cracks all over and we've seen all kinds of bugs in here, so we clean and spray quite a bit. (mostly ants and spiders, yuk!)
My son also has minor allergies, so I also give him over the counter meds for this and any reaction to bug bites, non drowsy, so he can play outside all day, without allergies effecting him, or bug bites (with the bracelet, when he remembers!)