bee-bow'-er (with a short "o", like the bow of a ship)
Very close to jbsrawat's "bee-baw-er," and i agree with his(?) definition of "bower" as "a pleasant place in the shade under a tree, especially in a garden."
However, the first definition to be found in the Oxford English Dictionary is "A dwelling, habitation, abode. In early use literally A cottage; in later use a poetical word for `abode'," with a history of usage going back to Beowulf (pre-1000).
This definition suggests something along the lines of a (large) bee-hive (I'm thinking of the large, conical ones made out of woven or coiled sprigs).
Curiously, however, the word "beebower" itself doesn't appear in the O.E.D. at all, and a quick goggle.com search showed 996 hits, all of which seem to be a proper Surname (which I've never seen or heard of before).
The second sense of "bower" in the OED is "[Obsolete] A maker of bows; a bowyer," or "One who plays with a bow on a violin or other stringed instrument."
In which case the pronunciation would be "bow'-er," with a long "o."
But a "bee-bower" would make no sense in either one of these cases, as far as i can see.
Finally, the *fifth* OED definition opens up even better possibilities :
"[Obsolete] [from the Dutch _bouwer_ or Ger. _bauer_] A peasant, husbandman."
A "beebower" would, therefore, be a "husbandman of bees," nowadays a "bee keeper."
That makes the best sense, to me and explains the surname (which "bower" in the sense of a "hive" doesn't).
The OED doesn't suggest any pronunciation for this sense of the word, the last known appearance of it being in 1563. But surely it is very close to the German "bauer" to which it is related --i.e., a short "o."
Related to this "husbandman" sense is yet another : "A tenant who rents a herd of cows along with their pasture and fodder from a proprietor or farmer, and makes what profit he can out of their produce, after paying the rent; or who gives his labour as his share, and divides profits with the proprietor of the stock."
However, the OED only knows this sense in relation to cows, not bees.
So : "bee - bau' - er" or "bow," with a short "o."
Or, find someone with the surname and ask *her.*