Well, to muse is to think deeply about something. You can lie on your bed musing about a question, then you decide to go on Musing and ask it for real.

The word is also related to "muse". Muses are people or things that bring inspiration onto others.

Those two seem like pretty big candidates for the origin of the name, but we'd have to ask the creators themselves to be sure. :) For me, it just sounds pretty and is original, so it's a good name.