The apostrophe goes after the entire noun phrase that the possession relates to, which is one of my friends. This of course refers to a single individual (one friend), which is what you mean.
So for that reason, One of my friends' name is opposite to his personality is a correct written form. However, it doesn't sound very good like that, which is why we suggest changing it to The name of one of my friends is opposite to his personality. Notice that when written in this way no possessive apostrophe is needed, because of is used.