All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
The name Benyaameen (Benjamin) is among the non-Arabic names and naming with such names is permissible provided that they do not include a forbidden meaning. However, it is well-known that it is only the non-Muslims who name with the name "Benjamin", so it is not permissible for a Muslim to name his son with this name because it is imitating the non-Muslims.
What is apparent from the Fatwa of the Permanent Committee of Iftaa’ in Saudi Arabia in an answer to a question about calling a Muslim by Benjamin, Peter and John is that they (the members of the Committee) see that such names are peculiar to the non-Muslims. Their answer reads: “This act (calling by such names) is not permissible because it is imitating the non-Muslims in their names, and the Prophet said: "Whoever imitates a people is one of them.” Therefore, it is an obligation to avoid calling by these foreign names, even if it is done jokingly.”
If this is the case for just calling someone by these names, so what about naming with them?
Indeed, a Muslim should choose for his son a good name such as names which mean being a servant of Allaah [like ‘Abdullaah and ‘Abdurrahmaan] and the names of the Prophets and the like. This is better than resorting to naming with such names which a person does not feel comfortable about.
Allaah Knows best.