Why did so many 50s and 60s songs begin with the word “well”?
Well…why?
Filler text.
“Baby” still seems to be popular as well, if Justin Bieber is anything to go by.
If so many did, it shouldn’t be hard for you to give us a few examples.
“Party Doll” by Buddy Knox: “Well all I want is a party doll…”
“Be-Bop-A-Lula” by Gene Vincent: “Well, be-bop-a-lula, she’s my baby…”
“Heartbreak Hotel” by Elvis: “Well, since my baby left me…”
Longstanding tradition of English-language narrative to begin with a word of pause as the speaker gathers his thoughts and prepares to orate, going back to Old English if not longer. The first world of Beowulf is “Hwaet…” which translates to “So…”
Deep subject 
< runs away>Ok, a few:
“Well hello Peggy Sue…”
“Well she was just seventeen…”
“Well no one told me about her…”
“Well shake it up baby now…”
“Well, they took some honey from the tree…”
Paul McCartney’s song “Long Haired Lady” starts with 5 "well"s. Can anybody beat that?
“Well” is a common way to start an anecdote that you’ve been asked to tell…
“Hey Hank, I heard you had a crazy night. Tell me all about…”
“Well…”
The majority of The Beach Boys catalog. 
Don’t poison the thread.
So help you Rhonda!
All too many people (including Eddie Vedder) think the chorus of “Last Kiss” starts “Oh where, oh where” But it’s really “Well where, oh where”. Cite.
It doesn’t start with well, but John Lennon’s “Well Well Well” contains a lot of wells.
Yes, and I guess I have assumed that early rock got it from the blues.
Well I woke up this morning to find my baby gone…
Well I woke up this morning and I got myself a beer…
That type of thing…