Interesting question. We know about the other planets of our Solar System; even ancient peoples knew about the ones that can be seen with the naked eye. But we have no reason to think that Eru Ilúvatar would have populated any of them. Tolkien never even set down any legends about the Moon being inhabited. He did set forth the legend that the planet Venus was the light of the Star of Eärendil, the Silmaril that was recovered from Angband by Beren & Lúthien and later given by Elwing to Eärendil.
Well, no story from Middle Earth has The Moon inhabited, but Roverandom does.
Also, there are the two poems from The Adventures of Tom Bombadil, The Man in the Moon stayed up too late and The Man in the Moon came down too soon, which suggest that either a Man does live on the Moon, or that Bilbo was really creative.
It's probably from a Hobbitish folktale or a Mannish one that the Hobbits adopted.
Fans have speculated on other planets in the solar system:
Tolkien wrote a couple of Hobbit poems about the Man on the Moon. Bilbo wrote one of them.
The Man in the Moon Came Down Too Soon
The Man in the Moon Stayed up Too Late
In Elvish mythology, Tilion the Hunter is the Maia who steers the moon. Early drafts had other spirits assisting him, but these Maiar were dropped in the published Silmarillion.
Arda: *has cool life and waters and air currents and landscapes and caverns and civilizations*
Tilion: Guess who's goin to the moon
I always liked that the song that Frodo sings in the common room of the Prancing Pony is a take on the old nursery rhyme "Hey DIddle Diddle". It's as though Tolkien was saying that this was the origin of the nursery rhyme.
Somewhere Tolkien did say that it was probable that "Hey diddle diddle" was taken from Bilbo's poem. (He says it like Bilbo was part of history, that is pretty cool) :)
In Tolkien's universe, the Moon was the last flower of Telperion, as the Sun was the last fruit of Laurelin. Tilion was a bit wayward in his path and steered too close to the Sun, being enamored of Arien, the maiden steering the Sun. The Moon was then scorched. This would explain its phases.
In his book The Hobbit Companion, David Day explains that Bilbo means short sword or rapier, from the Portuguese town Balboa. He does go on to say that Bilbo's sharpness is that of his wits rather than his blade.