The DC Silver Age superheros appeared to have teenage sidekicks almost at random. Batman did certainly, Superman did sort of, but Green Lantern, Atom and Hawkman all eschewed the buddy combination, as did later DC heroes like the Spectre, Metamorpho and the Creeper.
But Flash got Kid Flash, and since it became so significant an event in the DC Universe, it is worth looking back on. Kid Flash first appeared in Flash #110. Wally is Iris West's nephew and Barry feels an affinity for the young lad, who's clearly one of the Flash's biggest fans. But when the Flash appears himself (of course Barry slipping away), lightning does strike twice.
I suspect what happened was a suggestion from the publisher for a Kid Flash character, because that is one of the lamest origins for a DC character ever. Oh, by a wild coincidence he got hit while by lightning standing in front of some beakers and test tubes, just as Barry did? That's lame and intentionally so.
Like all of DC's young heroes, he starts out as a boy and quickly becomes a teen. In Flash #110 he looks to be about 8 years old; by the end of the 1960s he will be clearly a man.