Grandpa plays the harmonica as a hobby—and since hobbies are the thing Grandpa is most serious about, the harmonica is a kind of symbol of his essence. And what is Grandpa's essence? The idea that you shouldn't work too hard. Play the harmonica instead!
In the play, the harmonica becomes a symbol too of Mr. Kirby's lost, more cheerful past, when he played the harmonica and had a friend or two. Grandpa gives Kirby his own harmonica when the two are in prison, and Kirby takes it out and fiddles with it while he's closing his big business deal, and wondering if all the money is worth it. ("It isn't," honks the harmonica.)
Finally, at the end of the play, Grandpa and Kirby play a harmonica duet, which prompts Tony and Alice to reconcile and everything to turn out swimmingly. Harmonicas; they're magical. (Also somewhat heavy-handed. But nobody said Frank Capra was subtle. Toot.)