The correct answer is: I went with my cousin, whom we all like.
The relative pronoun here is whom.
A relative pronoun is used to connect a clause or a phrase to a noun or pronoun. The clause modifiers or describes the noun.
Whom refers to my cousin and describes him : whom we all like ( we all like him ). The clause describing the subject (my cousin) is called the relative clause. It is always introduced by a relative pronoun such as: who, whom, which, that.
