I agree with thegiffon on Allowin.
I think a connection of Begga with BEK is unlikely, because bek has a /k/ sound in low German (and is cognate to High German Bach). I'm tempted to postulate a connection to the modern English word beggar and the Dutch words Beghard and Beguine (two mendicant orders); but their etymology is unclear and they are only attested from the 12th century on. And than there is St. Lambert le Bègue (Lambert the stammerer).
Another more conventional way of thinking is a connection to the name element BERG, BURG with loss of the /r/ sound (no unattesteted, as the connection of Fed ~ Fried shows).
All in all, there is no really satisfactory explanation for the name Begga and we should leave it as "unexplained" for now.
--elbowin