Question

What is the direct object in the sentence? my neighbor mode has a lot even though it was raining

Answer 1

I feel like raining is your direct object. 

A direct object forms from the action performed by the subject. 
Your subject is either your neighbor or whatever was supposed to be after a lot. 

Answer 2

The direct object in this very odd sentence is lot.
What does my neighbor mode have? - A lot.

To the person above me - raining cannot possibly be a direct object because it is a verb.