There are a few things you can do. You could look for a place that has a
McNally Strum Stick
Which sounds good no matter what you strum. Although it is a little longer then a ukulele, it is really small and kids love it.
Or get a ukulele and learn how to tune it so that it is "an open tuning"-- basically, so that the strings are all tunes to play a note even if they aren't pushed down on on the fret board, if that makes sense? The beauty of guitars/ukuleles are that you can tune them ANY way you want!
The strum stick is really cool and our kids love it. Please don't waste your money on the "kiddy" guitars. They usually really bad, all apart and discourage kids from play because they are made so poorly. If you do buy a real instrument, just show your child how to hold it, and try to not let them walk/run with it. We have a special place in our house where the instruments can be played, they can be moved CAREFULLY are just work hard to train the kids to always sit down before playing. Sometimes they want to rock out and need to stand, so a shoulder strap that actually fits them is a good idea, but watchout for going through doors.
But just like anything, kids learn to treat them with respect and love. See if there a is music store with a strum stick nearby... they're totally fun. (it is also known as a mountain dulcimer. )