Calling "Bikini" the hot ham-and-cheese sandwich is not an exclusive characteristic from Barcelona but from the whole Catalonia, but the origin of the word is fully Barcelonian.
All the locals know the Bikini concert hall, in the district of Les Corts. Well, Bikini opened in 1953 as a dance hall with terrace and mini golf, and also as a meeting place for dining. Among its culinary specialities there was an adaptation of a well-known French sandwich, called croque-monsieur. This snack was becoming increasingly popular and was eventually known as "the sandwich they prepare at Bikini". From that to the current name of the sandwich there was only one step.
Well, or at least that's what I've heard and what I've been able to investigate. If anyone is aware if that is an urban legend, please tell me!
What it is for sure is that I know more than one person who has asked for a Bikini at a bar outside Catalonia and has been looked at as a freak.