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This surname appears to be restricted to the USA and rare even there. There is a similar name, Awsby, perhaps the same in origin. This last name looks like an English place name and perhaps derives from the Cumbrian villages of Great and Little Asby. The meaning of this name is "ash (tree) farm".
I believe the place name Asby is also found in Scandinavia, with the same meaning. English place names ending in -by are of Danish or Norse origin.