Significance of the green stone?

[Firefoot]

In the Chapter "Many Meetings" in FotR, Bilbo writes the song/poem about Eärendil, and afterwards, Bilbo says to Frodo:

In fact it was all mine. Except that Aragorn insisted on my putting in a green stone. He seemed to think it important. I don’t know why.

Is there some special significance to the green stone? I don’t think Tolkien would draw attention to it if there wasn’t.