So I mentioned Peter Cook as the priest in The Princess Bridge in another thread and it got me thinking. That has to be the most influential minor role in movie history. It was really small role, just a handful of lines in a single scene. But it is one of most memorable roles (in one of the most memorable and quoted movies of all time, so that’s saying something). I cannot count the number of times I had it quoted at me over the course of getting married in the UK (at an Anglican church)
What other contenders are there? I don’t mean just “not the main character”, I mean a really minor role with handful of lines in one or two scenes.
Benicio del Toro in The Usual Suspects? It may not be quite as minor as you’re looking for, but I would submit that it would seem minor indeed if del Toro hadn’t knocked it out of the park.
Q usually only had short screen time in the James Bond movies. Usually only one scene where Bond is given his equipment, and sometimes another shorter scene at the end. But they were always memorable.
John Turturro as the Purple Jesus. No influence over the story, but most definitely memorable. Especially when he licks the bowling ball.
Tara Reid as Bunny Lebowski. Not much screen time, but a big chunk of the story revolves around her character, so she’s certainly influential. Maybe that’s cheating, though, since any character fulfilling the same function in any crime-based movie would influence the story. Still, the line “I’ll suck your cock for a thousand dollars” is priceless (even though she quotes a price for the service).
Both Beatrice Straight and Ned Beatty were nominated for Best Supporting Actress/Actor for their roles in Network despite Straight only having five minutes of screentime and Beatty six minutes.
It’s a slightly different question, but “best character that had no role in the plot” is an interesting criteria. It is a delicate thing to have a whole bit onvoling a character that could be removed without changing the story. It canbe valid, of course, because it goes to character development or exposition. But typically in those cases you don’t want the superfluous characters to be stealing the show. But sometimes it works.
The OP wouldn’t meet this requirement because the priest being slow and fumbling and clueless is a major part of the plot: its why Buttercup never had to say “I do”. But some of the others probably do.