It was revealed in the last episode.
When he was born, Wally could not pronounce Theodore and instead called him "Teeder". Ward and June changed this to "The Beaver".

From his 2 front teeth.

Theodore Beaver

And to add to what everyone else said there is the most obvious reason that he got the nickname Beaver. It rhymes with Cleaver.
I would say it's not that unusual for a bunch of kids to give another kid a nickname that rhymes with his last name.
(knock,knock,knock) Penny (knock,knock,knock) Penny (knock,knock,knock) Penny

I'm not sure why he thinks that was a lame explanation for the name. That's the exact way my older sister got her nickname. 50 years later we still call her that.

Thank younow it makes more sense!
For the longest time, I thought it was because of his front teeth but then I noticed Wally's 2 front teeth were kind of big too.
I had the chance to work with Michael Jackson who was as brilliant as they come.
Tommy Mottola

"When Beaver used to get nervous before a test, he would chew up his pencils. One time Ward said, 'If you keep that up, you'll turn into a beaver,'
That does sound more plausible.
I had the chance to work with Michael Jackson who was as brilliant as they come.
Tommy Mottola

But I really can't see how a mispronunciation of Theodore comes out as Beaver.
I always thought it was a dumb explanation and felt the writers should have taken a few more minutes to come up with something more plausible, like, "When Beaver used to get nervous before a test, he would chew up his pencils. One time Ward said, 'If you keep that up, you'll turn into a beaver,' and that's when the name stuck!"
Emojis=
Emoticons=

Thank You!
I thought I had seen all the early episodes. It is one of those little things that aren't really important but . can't get it out of your mind. LOL
But I really can't see how a mispronunciation of Theodore comes out as Beaver.
I had the chance to work with Michael Jackson who was as brilliant as they come.
Tommy Mottola

It was explained in the last episode that Wally Could not pronounce Theodore
correctly when his baby brother arrived and it sounded like Beaver & the name stuck