On Leave it to Beaver get that “nick” name?

That's the actual quote.

Here's what most people think it was: "Ward, don't you think you were a little hard on the Beaver last night?"

(I don't know how he came to be known as The Beaver)

It came out theater, modified to Beaver

then he dubbed his own voice in for the English language release.

He had a great voice.

(It was a female)

Did Drew Barrymore ever host when she was a kid, maybe?

She was seven

That's a lot younger than I would have thought. My next guess would be Dakota Fanning.

and what was Mr. Peel's first name.

What was the original name of the show?