This name comes from the 19th and 20th century innovation of using jewels as feminine names. The word 'Amber' does not appear until the 1600s. Confusion subscribing this name to the Medieval period may have come from the Victorian habit of giving their children medieval names which coincided with the first known appearance of Amber as a given name. Also, the surname 'Amberson' which has been around for much longer was mistakenly believed to be 'son of Amber'. It's true meaning is 'son of Amery'.
"Amber" also is a color. Like the sap, it's a mixture of orange and brown.
― Anonymous User 4/19/2006
It is possible that "Amber" (the stone and the name) 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". Interestingly, "umbra" has lent its name to two colors: amber (yellow) and umber (brown).
Everyone keeps saying that Amber is fossilized tree sap, but that's technically not true. It's actually fossilized tree resin. Resin & sap are not the same thing; not that one is necessarily any better or worse than the other. Anyways, just thought some people might be interested to know that.