What's the correct preposition to complete "to be baffled ... why"?

Solution 1:

All OP's alternatives are acceptable except on (over isn't good, but it's not totally invalid to me).

My vote for the "best" would be "baffled as to why", which Google Books says has 2350 hits for C21, against 1180 for plain baffled why. The others are ..about;301 ..at;143 ..by;232 ..on;8 ..over;31

Per comments below, baffled by accounts for a good proportion of all usages for baffle, but it seems we tend to avoid the alliteration of baffled by why.

I myself am baffled by the etymology of baffle, which seems to have originally meant disgrace, thence through mock to the modern sense of to bewilder, perplex. I also think we tend to use the "passive" past participle more today - we say "He's baffled by it", rather than "It baffles him".