I entered a name that I really liked, but it said it was rejected. I left it as was. The next day, I logged on, and it was accepted! Try!

It will often say it is rejected immediately if you accidently enter it more than once, because it sees your prior attempt and considers at as name already in use.

Guild names seem to be actually reviewed evey 24-48 hours, where pirate names seem to be reviewed every few hours.

I only entered it once. ?

You must have hit enter twice or something. Or the rejection you saw was pertaining to prior request. They cannot 'reject' immediately other than as a duplicate (or if it has something inappropriate from thier filters, but that would NEVER be approved)

It the name a person wants is rejected, they can request it again. It will often be accepted on the 2nd or 3rd try.