A lot of times these commercial instruments for children will be constructed with very little emphasis on pitch accuracy. Walking through toy stores, I've found some that play some pretty other-worldly scale patterns. Bearing that in mind...
Assuming this thing starts on a tonic and goes in a conventional diatonic pattern upward to the sixth, then as American Luke mentioned, you'd be stuck with playing music that:
Does not span an interval greater than a major 6th, AND
Does not use any tones outside of the diatonic major scale between the tonic and 6th.
Let's say the bottom note is a concert "C" pitch. In the perfect world, your alligator xylophone would consist of these pitches:
C, D, E, F, G, A
A lot of children's songs fall within the above guidelines. Off of the top of my head I can think of these:
Mary Had A Little Lamb
Itsy Bitsy Spider (Starting on the high "G" rather than the nonexistent low "G")
Row Row Row Your Boat (Omitting the high "C"s on the first occurrence of the 'Merrily' lyric)
Twinkle Twinkle Little Star / ABC Song
Farmer In The Dell (Again, exchanging the high "G" for the first note)
Frere Jacques (Once again, exchanging the high "G" on the last phrase 'Ding-Dang-Dong')
Rain Rain Go Away
Camptown Races
Humpty Dumpty
London Bridge Is Falling Down
Pop Goes The Weasel (High "G" for the first note)
Oh Susannah
This Old Man
By no means is this a complete list, but at least it will give you a decent starting point. A rule of thumb you can set for yourself is:
Try playing the song on the xylophone.
If the melody has notes that don't exist on the xylophone, throw it out.
If the melody has notes that DO exist but have to be played down/up an octave, adjust the melody to play within the notes on your xylophone.
If you follow these simple rules, you and your family can assuredly have a fun and memorable alligator xylophone experience for years to come! Good luck!