If you want sun protection you need a wide brim and a pale color. A dark hat will make your head so hot that you'll want to take it off.
I use a Watership Trading Companie Vineyard Haven hat. Not cheap, but worth every penny to this freckled redhead. It would be a bit too much of an investment for a growing kid, though.
My kids wear bucket hats when they are not trying to mooch my VH. Neither one of them will wear a sun hat anywhere else, but at FL theme parks they beg for them. Getting a bucket hat to be autographed is fun for a lot of kids -- use a laundry marker to get the best washable result.
If a ball cap is the only thing they will wear, go to a fishing store like Bass Pro and get one with a tuckable neck shield.