From what I understand, the design itself is based on a long-ago conversation about Falo needing a symbol, and someone looked up at the sky, and it was a cloudy night with a full moon. (I'm sure if this is inaccurate someone will be kind enough to set the record straight.)
This night sky was turned into a symbol, which is fairly blobby looking in most cases, and then, when a new moonstone design was needed, it was burned onto wood.
My first reaction to our medallions is, "What could possibly be period about wearing a nametag?" My second is, "Ew, it has F.A.L.O. on it." The name of our group is not medieval fantasy in any way (despite a noble effort at revisionist history to make it an elven word), and tends to jerk me back into the modern world when I see it. A fair number of those so honored don't openly wear them, although I can't speak for their reasons.
However, tradition does not need a solid foundation to exist, and we've had the same design for a long time. The symbol, nondescript as it is, has probably been linked to our group by many.
I've made no bones about not caring for the design, but I offer a compromise: keep the basic design (symbol and stone on wood), but change the writing. Remove the FALO, or change it to elven or other cool-looking characters. Replace the member's name with a sigil to represent them - as an example, take a look at how Ellam writes his name on his artwork. A sigil combines letters of a name into a single symbol of great magical power. That would also reinforce the idea that moonstones have power.
This suggestion would address a lot of the concerns of those who want to change it and preserve a lot of its character for those who would like to see it stay the same.