Should this object plurality agree with the subject plurality?

In this sentence, (a) what part of speech is "name", and (b) should the plurality agree with the subject? (i.e., should it be "name" or "names"?)

• The first answer is given by the names that we choose for ....
• The first answer is given by the name that we choose for ....

a) "name" is a noun in your sentence.

b) The plurality of "name" depends on the situation you are trying to describe. If you only choose one name, then "name". If you choose multiple names, then "names". The plurality of "name" does not have to agree with the verb "is" since it is not the subject of your sentence (subject is "The first answer").