If I may chime in.
I love "Leave It to Beaver" dearly, but I believe "Father Knows Best" was the better show in every way.
One of the great things about LITB is that it dealt with the small, everyday "problems" that could happen to anyone -- and I believe it might be unique among all sitcoms in never having a storyline that could make you think, "Oh, that could never happen."
But "Father Knows Best" often dealt with bigger issues, or seemed to be "bigger", in the way the show was acted, written and directed. So I think "Beaver" seems a little trivial compared to it.
In her Archive of American Television interview, Elinor Donahue said that Robert Young's production partner, Eugene Rodney, kept very tight control over the quality of the series, literally going over every word of the script. "Father Knows Best" was one show that never "jumped the shark" -- if anything, it seemed to get better each season.
Also, "Father" was more successful in every way (at least, originally). "Beaver" never cracked the Top 30, and "Father" was #6 in the ratings when Robert Young decided not to do the show any more. "Father" was nominated for and won several Emmys, and "Beaver" had no nominations after its first year. Also, "Father" was on the cover of TV Guide many times, and "Beaver" only once.
Of course, in the long run, it's possible that "Beaver" is the more successful and better-remembered, due to being a bigger success in syndication.
I'd truly hate to make the decision, but if I had to take only one show to a desert island, I'd take "Father" over "Beaver."