Now that just about every harp has been recommended for a child's first harp, I thought I'd add my $0.02.
When I was five, my dad came home with three harmonicas one day, I think it was a blues harp, an American Ace, and a Marine Band, I think they were all in C. He had been listening to all kinds of blues, but especially loved the harmonica blues of Sonny Terry and of Paul Butterfield, both of home were out in L.A. from time to time. He kept the Blues Harp (that's the one I'm not totally sure of), gave me the Marine Band, and gave my brother the American Ace (I was jealous because of the picture of the airplane on the American Ace, versus mine which had that teensy tiny picture of the Marine Band). He gave us the harmonicas and put on familiar Sonny Terry record and told us to make our harmonicas sound like that. It was harder than I expected it would be.
Anyway, of the three of us, I'm the only one who went anywhere with the harmonica. Now maybe if he'd given my brother the Marine Band...