Performance Today fans already know about Bruce Adolphe's Piano Puzzlers. (Bruce takes a simple and familiar melody, and re-writes it in the style of a famous composer.)
I've stumbled across a few other musical puzzles online. Conductor Nicolas McGegan has double acrostics on his web site. The answer to the most recent puzzle was an amusingly dated concert review: "Lovers of Brahms were much disturbed by large numbers of people leaving the hall between movements of the C-minor Symphony. It must be admitted that to the larger part of our public Brahms is still an incomprehensible terror."
Not sure when the new puzzle went up, but McGegan says the first ten to submit correct answers get a prize. And McGegan gives a couple of hints...the answer is another concert review, and the name Berlioz appears in the url.
Here's a nice online version of musical puzzles by Bach, his "puzzle canons" Gotta know your stuff to tackle these, but just reading the answers gives you a clue how much thought Bach put into creating musical subtext, not just beautiful sound.
And of course, there's always Elgar's Enigma to fall back on.