Hi Kate,

This may give you some leads ...

"Bangle" surname ...

Altered spelling of German Bangel, which probably is a variant of Bang or of Bengel, or derives from a field name in the district of Lüneburg.

"Bang" surname ...

1.Korean: variant of Pang.
2.Norwegian: habitational name for someone who lived at a farm named Bang, from bank ‘flat hill-top’, ‘terrace’.
3.Danish: from Old Danish bang ‘noise’, hence a nickname for a loud or brash person.
4.German: nickname for a timid person, from Middle High German, Middle Low German bang ‘fearful’, ‘nervous’.

"Bengel" surname ...

1. Habitational name from a place so named, for example in Bavaria and the Mosel area.
2. From Middle High German bengel ‘club’, ‘cudgel’ or in a transferred sense ‘hooligan’, ‘lout’, hence a nickname for a boorish man.

Source:
Dictionary of American Family Names
Oxford University Press
ISBN 0-19-508137-4

The fact that this book has "American" family names in its' title I don't think has much meaning when researching surnames. (Nothing specific to America.)

If you key in the Bangle surname at www.familysearch.org
that can also give some indication of the localities of the surname - and variants.