I have to go with the 1961 episode “In The Soup”, in which Whitey utters the immortal words:
Eddie has a good line later on:
My favorite moments would be Anything with Gus, The Fireman. When things were REALLY tough, Beav could Always go to Gus for the Straight Dope.
Any appearance by that closeted queen Richard Deacon is the highlight of any episode. 
My favorite Beaver moment isn’t a quote at all. In one of the earlier episodes (I don’t even know which one, but the Beav was quite small so it had to have been the first or second season), Beaver comes running in while June is one the phone. She motions him to be quiet, and he comes up to her and plays with the phone cord until she swats his hands away. Leave It To Beaver has a reputation for being unrealistic, but that scene was one of the most realistic portrayals of what young kids are like that I’ve ever seen on TV, or in a movie.
One episode, Beaver had a school friend whose family had immigrated from a Spanish-speaking country and who didn’t speak much English.
Eddie (who had taken Spanish in school) smerked to Wally that they should teach Beaver a phrase meaning, “You have a face like a pig” and claim that it means, “You’re a swell guy”.
Wally exploded, telling him, “Eddie, that’s a dirty trick!”
Eddie replied: “My dad says nothing’s a dirty trick if it’s funny!”
That one line summarized all you needed to know about Eddie, and succinctly explained why he was so messed up.
I loved how Mr. Cleaver was so diplomatic about being ‘on’ to Eddie, without ever letting on that he was. He once remarked to June, “That boy’s so polite, it’s almost Un-American”.
Eddie was the high point of that show for me.
Usted tiene un cara como puerco (sp?)- God that was a hilarious episode! I also love the whole retro-ness it has now, like when Ward tells Wally to put on his good suit because they are going to a football or basketball game- can you imagine that now?
"And I’m not never comin’ back, " Beaver says forlornely as he prepares to run away.
I don’t remember the exact quote, but there was a comment by Ward to June (Wally had disobeyed him about something, but it turned out he had a good reason) that went something like, “You know, before we had kids, I never thought I could be furious at someone, proud of them, and utterly confused by them all at the same time.”
I loved when they invited the boy’s parents to come over so they could sort out what had happened, and Ward asked Beaver to repeat the offending phrase.
When the parents huffily bolted, June asked, “Beaver, what does that mean?”
He replied, “I don’t know, but whenever I say it, everybody leaves the room!”
I was never a Beaver gal (!), I was more into The Donna Reed Show (Donna was a goddess). So, tell me, is the old wheeze about, “Ward, you were a little hard on the Beaver last night” an urban legend, as I assume?
So far as I remember it was, Eve. I haven’t researched it, but I’ve seen every epidsode of the show several times, and I don’t remember that line ever being spoken. I like The Donna Reed Show, too, and Father Knows Best, too. I was never an Ozzie & Harriet fan, though.
As the show went on, and Eddie grew more obnoxious and obsequious, June developed a sort of eye-rolling, just-stepped-in-dog-poo expression. It was probably Barbara Billingsley’s greatest acting trick until Airplane!