Neither of my friends ____ agreed to go with me.
Neither refers to 'no one' which is singular. Therefore, it is 'has' the correct answer.
Good to see that majority of voters do know the correct use!
'Has' is the third person singular, so how can the word be used to describe what 'friends' (plural) will or would do? In this case it can't be assumed that 'neither' is the same as 'no-one'. the two are entirely different. So, 'neither of my friends agreed to go with me' is fine and so would 'one of my friends has agreed to go with me'. No-one, is nobody (singular), whereas 'Neither' refers to 'not this/that one thing (or person), or this/that one (thing (or person).
"Neither" means "neither one", as in 'neither one of my friends', which is singular. Correct usage is "has".