I'm going way back here...I have a Williams ancestor who came to MA on the John and Dorothy in 1637. On the ships list his wife was only listed as Ales. Traveling with them was Katheren Raby and early genealogists believed her to be Ales' mother. Katheren was a seaman's widow.
In trying to find them I have found (IGI) a Catheren Thomson who m. a Barnaby Raby in 1592. Saint Peter Parmentergate, Norwich, Norfolk, England. Coincidentally an Alce Bebe was baptised 1600. I could find no other Bebe's listed at that time and wondered if it was another alternate spelling. I've seen a bunch in the old records!Usually the soundex finds the variations. I found the Alce Bebe by searching on the Ales within the parish records.
Example- the Williams family moved to Salem MA and there is a Catheren Rabbe there!
Perhaps some of your early family records can be found in that parish. Hope it helps.
If the Bebe name was prevalent in that area, please let me know and I'll rule out that possibility. Thanks.