A reason for saying that “God only knows” is that it would be impious or irreverent to suggest that He (who knows everything) does not know.

For that reason, you get [expressions like this one on the Tomb of the Unknown Soldier](http://english.stackexchange.com/q/9235/9548).

A corresponding secular phrase could be simply, "I don't know", "No-one knows", "It's unknown", or "It's a mystery".