Remember a few years ago I did a cartoon of Dennis shouting up the chimney his thanks to Santa for the sugar tongs? You could have a Beau Peep cartoon be the Christmas competition.
Or there is always the old standby of guessing how many marbles are in a jar. To make it more christmasy you could change the marbles for salt water taffy.
Or you could ask us to assign the characters of Beau Peep into parts in a modern famous movie or classic novel. e.g Mad Pierre could have the title character spot in Frankenstein.
Or people could print out a Beau Peep face and make a cut-out and have their photo taken with it in the most unusual place.
Or there could be a make a Beau Peep board game competition.
Or there could be: make a classic Beau Peep dish inspired by Egon and take a photo of it. Listing the ingredients used.
How about a theme song? I can't write music though so you would have to allow dum de dum de dums.