How can a girl convince her mother to agree to her marriage to her paternal cousin, who is a religious and well-mannered young man, given that the disagreements are limited to those between the mother and the young man's mother only?

You must obey your mother and not marry your cousin except with her consent. To solve the problem, it is advisable to turn to Allah in prayer, to tell your mother frankly about your desire and explain the young man's character and religiosity, to seek help from someone who has influence over her to persuade her, and to strive for reconciliation between your mother and your cousin's mother. You must also be patient and content with Allah's decree, for there may be good in disapproval, as Allah Almighty said: (And it may be that you dislike a thing which is good for you and that you like a thing which is bad for you. And Allah knows, while you know not).