Ambra

It is possible that "Ambra" is related to "Umbra" which means in Latin, "shade" or "shadow", and has come to mean archaically in English, "phantom, ghost", "an uninvited guest accompanying an invited one" and in astronomy, "the shadow cast by the earth or moon during an eclipse". Umbra has given rise to such words as, "umbrageous", "umbrage", "umbrella", "penumbra" and "somber".

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Interestingly, "umbra" has lent its name to two colors: amber (yellow) and umber (brown), so there may not be a direct connection from "amber" the semi-precious stone to "Amber" the popular girl's name, but rather, "amber" the color may be the progenitor of both the stone and the name. It just follows, it seems to me, that Ambra, the girl's name, also shares its roots with Amber, possessing the same lineage.

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It's a stretch but worth mentioning that the Arab-Hispanic place name, Alhambra, is said to mean “the red palace”. Possibly, "Ambra" can find its roots in Arabic OR the word "red" in Arabic might share its roots with the color names, "amber" and "umber". I hope an Arab-speaker comes along and clears this up eventually.

Ambra is also a feminine version of the Norwegian name Ambros which was most common in the late 16th/early 17th Century. It has nothing to do with the jewel Amber, a word which does not seem to appear anywhere before the 1600s