What was the name that King Benjamin gave the people that is referred to in Mosiah 1:12?
The name that King Benjamin gave his subjects, as referred to in Mosiah 1:12, is the name of Christ. If you will notice in verse 11,
And moreover, I shall give this people a name, that thereby they may be distinguished above all the people which the Lord God hath brought out of the land of Jerusalem; and this I do because they have been a diligent people in keeping the commandments of the Lord (Mosiah 1:11)
that the footnote by the word ‘name’ leads you to Mosiah 5:8-9 that specifically answers your question.
And under this head ye are made free, and there is no other head whereby ye can be made free. There is no other name given whereby salvation cometh; therefore, I would that ye should take upon you the name of Christ, all you that have entered into the covenant with God that ye should be obedient unto the end of your lives.
And it shall come to pass that whosoever doeth this shall be found at the right hand of God, for he shall know the name by which he is called; for he shall be called by the name of Christ. (Mosiah 5:8-9)