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I have been reading the English surname books, and they all give equally vague and uncertain descriptions of Asbell, Azbell, and Isbell. One webpage on Azbell gives an interesting theory for the lack of documentation for that name prior to 1600, that the author believes the name is an alias for a Jewish or other family gone into hiding. Isn't it just as likely, even more plausible, that the name is a phonetic translation from French or Gaelic Irish or Welsh or Spanish, by one immigrating into England at a later time?
I wonder if the Asbells and Azbells might be a branch of the ISBELL family. The name is supposed to the Spanish spelling of Isabel. In France the name is Isabelle, and one supposedly came to England in or shortly after 1066, but details are sketchy. It is to be noted that the spelling "Isbell," apparent at least since the 1400s, is the Spanish spelling of the name.
If anyone has other theories to advance, please let me know.