There is no problem with using a word in prayers that is not in the Qur’ān, as long as the word is relevant and appropriate. Allāhu Akbar (God is great) is also not in the Qur’ān but is mentioned several times in prayers.
Amīn being the name of a person also does not change anything. It is the meaning of it (and not one who owns it as a name) that is intended when people use it.
Also, it is not necessarily correct to hold that something originated from Jews or Christians should not be used. Muslims, Christians and Jews are all followers of the religion of Abraham (sws) and have lots in common.
Āmīn simply means "so be it". It is a reasonable end to any supplication. We cannot categorise it as the Sunnah of the Prophet (sws) but we also cannot condemn its usage. It is a permissible word to say after any supplication simply because it is relevant. Of course we also have Aḥādith narrated from the Prophet (sws) suggesting that the Prophet (sws) encouraged people to say it after supplications. There are also a great number of traditions which tell us that the Prophet (sws) commanded the believers to say Ᾱmīn when the Imām completes reading sūrah Fātiḥah. He is reported to have said:
Abū Hurayrah reported: The Messenger of Allah (sws) said: Say Ᾱmīn when the Imam says Amīn, for it anyone's utterance of Ᾱmīn synchronises with that of the angels, he will be forgiven his past sins. (Muslim, No: 410)
Hope this helps.