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Bevan, Biven, Bivens, etc. are all derived from the early "ab Ieuan" i.e. "son of Ieuan". Ieuan is pronounced something like "Y-eye-an" and was itself derived from "Johannes", borrowed in the Roman period (i.e. before about 400 AD). Ieuan tended to become the forename Evan and in time became the surname Evans or Bevan (from ap Evan, or son of Evan).
Morgan & Morgan in "Welsh Surnames" (U. of Wales Press, 1985) offer a nice example of the transition stage in your own spelling. "David ap David ap Iven" was recorded in 1533.
It is interesting to note that a spelling like Bivens or Bevans will tend to indicate a family of Welsh origin whose surname only became finally fixed after settlement in England or, more typically, in America. Bevan (and the variations) were originally "ap Evan" and already had the meaning of "son of Evan". The final "s" in the Bevans / Bivens spellings would therefore have been illogical within Wales.