How old is the kid?
I've been through three kids worth of arrow building and I started out with 5/16 for my daughter when she was 3 (she's 13 now). What I ran into was poor trajectory and often the arrows would not stick into the target. I could see her frustration when she got up within about 10 feet of a target and sometimes the arrows still fell out of a the butt or bounced off a 3D. Almost didn't want to shoot any more.
The following summer I happened to find some 1/4" field points that looked just like grown up points but were made to fit 1/4" shafts with NO TAPER cut. Just slipped over the square cut end. I made up some 1/4" hardwood dowell arrows and WOW!!! What a difference!!! They flew more than twice as far and twice as fast and they hit the target with authority and stuck real well. Since then, I've really paid attention to kids shooting and I can pretty much say across the board, kids want their arrows to fly far and STICK!!! 1/4" hardware store dowels get my vote hands down. They are cheap, you can get them at walmart in the craft section or in Lowes or Home Depot for about 35 to 50 cents each. Pick through the bundle and look for basic straightness, lack of knots and grain runout. You won't find any that are straight I'll tell ya that right now. The best ones are straight for 1/2 to 3/4 their length and then maybe have a gradual bow. Some will be evenly bowed over the whole length. Either of this type will usually straighten easily and well. Avoid shafts with multiple bends in different directions or any bend that looks like a kink or "dog leg" unless it's near an end that will be cut off. Give them a little flex right there in the store to see how they straighten. If you run across any that bend like a piece of lead, don't buy it. I mean, you should have to work to straighten a shaft. Sometimes you find a shaft that has a bend and you flex it the other way and it just stays there (where you bent it to) like it had zero spring back. Don't bother buying it as it will never stay straight and will shoot like cr@p. You should easily be able to weed out a dozen or two shafts in about ten minutes.
By the way, look at the 5/16 or 3/8" dowells for shafts for yourself. They make great squirrel and rabbit flu-flus. Fletch them up with leftover scraps or a single full length feather spiral wrapped on the shaft and screw a 9/16 nut onto the tip and you have a very cheap and effective (disposable even) small game arrow.
One word of advice, NEVER cut a kids arrows to the correct length like you would for yourself. Always give them extra shaft so that they can safely pull their bow back to their ear or farther cause sure as $h!t the little stinker will try it at some point. Usually right about the time you figure all is ok and you look the other way... Saves on making new arrows for the next summer too since they have some room to grow into them.