I agree, that Frän would be prononced with an "a" as in "and" - I think it means "women" in some languages like dutch, and it might be a swedish name, because Ikea uses it for one of their products, and I believe they use first names a lot. :)
Fran would correspond to what your mother told you about it being a french name, but then the ä doesn't make much sense, since Fran in french is pronounced more like "Fron". I also don't think that "ä" ever existed in french ... so I also believe it's a creative spelling of Fran , maybe used in some nordic countries :)
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