She had a relationship with a man and wants to marry him, but her parents refuse and her mother is crying

Obeying ones parents is obligatory, as is respecting their wishes with regard to good things. It is better to respect their wishes in this matter, and Allah will compensate you with someone better. Your description of the man in your question, as having the necessary qualities that islam asks for: kafaa and "aql" plus he is a good beleiver, does not fit with the sin that he has committed by having this haraam relationship with you. Whether this relationship involved zinaa (illicit sexual contact) or not, the way out is to repent and show regret. If this relationship resulted in pregnancy, then marriage is not the solution to this problem, because in this case the child is for the bed (meaning he should be given his mothers surname and the zaani [man who committed zinaa] has nothing to do with him), as was reported from the Prophet (peace and blessings of Allah be upon him). If you both repent to Allah, and you manage to persuade your family , and your guardian agrees to let you marry this man, then praise be to Allah. The Prophet (peace and blessings of Allah be upon him) said: Any woman who gets married without her guardians permission, her marriage is invalid

(Reported and classed as hasan by al-Tirmidhi, 1021).

We ask Allah to help you to repent sincerely and to help you to do the right thing, whatever it may be. May Allah bless our Prophet Muhammad.