If my brother named John passed away and I meet someone named John, would I say:
Oh! my brother's name is John.
Oh! my brother's name was John.
Part of this answer will be influenced by your personal view of what happens to John after he dies. If you believe John is completely destroyed after death, then you would refer to him in the past tense just like anything else that no longer exists. If you believe he still exists in some form (i.e. his soul), then you could refer to him in the present tense, while referring to past (living) versions of him in the past sense (such as "John liked watching movies and was very active").
So if you say "My brother's name was John", you are implying that John no longer exists, that he changed his name, or that he no longer is called John because he died, while using "My brother's name is John" implies to anyone who knows of his death that you still think he exists in a form that uses the name John.
I presume we're supposed to refer to dead people in the past tense including their names. The moment that one ceases to exist means that anything to do with them should be referred to in the past. Saying my brother's name 'was' John doesn't mean his name has since changed since his demise but simply that he is no longer in our presence and therefore cannot have any attachment of presence to his entity.