If they state that you can only enter the contest once (even going so far as to send a personal message to a contestant, and very nicely request that they remove their multiple entries and only re-post their favourite), it would behoove you to ensure that your announced winner had only one entry.
Also, if that winner had more than one entry, maybe you should only consider their first?
And if you still decide to choose a winner's second or third entry, try not to edit their entry, and then declare it the best.
My final piece of advice is that if you still decide to accept multiple entries from the winner, and then edit the winning entry, PLEASE ensure that said winner isn't publicly listed as a FRIEND to the judges/organizers on Facebook.
Just a few shocking tips for the future. - QBF