Well, downs “no good when wet” knock is a little over-stated. You have to get a down bag pretty darned wet before its insulation goes totally south. But it can happen. And besides, in this case down probably is not going to fit the budget. The low end for a down bag rated to 0 degrees will typically run around $300, and it only goes up from there.
REI Polar Pod sleeping bag
Polar Pod sleeping bagREI makes a pretty decent synthetic sleeping bag thats rated to 0 degrees and sells for just $99 (rei.com). Called the Polar Pod +0, it uses non-name-brand polyester insulation and has a full hood, draft collar, and more. It isnt even all that heavy, at four pounds. But it is fairly bulky, so look into a compression stuff sack to ensure the bag packs down to a size somewhat smaller than your son. REI also makes a slightly spiffier bag called the Zenith 0 that goes for $169. Its slightly lighter than the Polar Pod, and uses an insulation that packs down a bit better and might last longer.
You might also look at the L.L. Bean Katahdin Climashield 0-degree bag ($139; llbean.com). It uses a proprietary synthetic fill that Bean says has one of the best warmth-to-weight ratios around. Its definitely lighter than most bags in this category, a feathery 3 pounds 7 ounces in the regular size.
Lastly, The North Face makes an attractive 0-degree bag called the Elkhorn 0° H.O.T. SL, which retails for $99 (but is currently on sale for $89 at campmor.com). Insulation is a typical polyester hollow-core fiber, but I do like the fact that it has a polyester shell. Polyester is better at shedding moisture than the more common nylon.
Keep in mind that sleeping bags alone dont ensure warm sleep. Your son will need a good set of long underwear to put on, including gloves and a hat. And with his low weight and probably high metabolosm, a bite of chocolate at bedtime would give him some extra heat-generating energy.