Should one abandon a negligent cousin who does not pray and does not respond to advice, or should kinship ties with him be maintained since he is a relative?

Your cousin is among the relatives whose ties of kinship are legislated to be maintained. Therefore, you must continue to advise him and call him to Allah by explaining the importance of prayer and the danger of abandoning it, and that one who abandons it is worse than an adulterer, a thief, or a murderer, and is even considered a disbeliever by some scholars. If he responds, then all praise is due to Allah. If he returns to abandoning it, then resume advising him and do not despair. As for boycotting (hajr), it depends on the benefit; if you know that boycotting him will deter him, lead him to repentance, and make him steadfast in prayer, then boycott him. Otherwise, do not.