I've had thoughts similar to the Might's connections with Radagast and his reported mastery of changes of shapes and hues, and they seem to make the most sense to me. Sometimes, though, I wonder if there is any similarity to the notion that Thorondor was a Maia who took the form of an Eagle, and from him the Great Eagles of Middle-earth descended (don't ask me to cite where in the HoME books I read this; I'm recovering from the flu, and it's a marvel my brain is functioning much at all
). If there is precedent for certain unusual but seemingly "normal" creatures in ME having this kind of origin, it's possible that Beorn had a distant ancestor descended from some Maia of Orome or Yavanna who inhabited ME during its early ages and favored the shape of a great bear, whose story we simply have never heard. This may be completely fever dreams, but it might point to a kind of explanation that Tolkien, alas, never got around to actually imagining, or writing.
That aside, I'm with the Radagast connection.