Either he or his friends_(have)done it?

Either he or his friends have done it.

Explanation:

The subject-verb agreement is an important rule that means the subject and verb must agree in number, that they both need to be singular or plural.  Sometimes it's easy to determine whether the verb should be singular or plural, but there are instances when it may be more difficult.

When it comes to the construction either...or, we need to be careful about the number of the noun/pronoun that precedes the verb. If the said (in this case) noun is singular, the verb will be singular, too. If the noun is plural, the same goes for the verb.

So, the verb have in the given sentence will be plural because it's preceded by a plural noun (friends). If it were the opposite, the sentence would look like this:

Either his friends or he has done it.