Her mother wants her to marry one suitor, but her father thinks that he is not suitable; what should she do?

Praise be to Allah.

Allah, may He be exalted, has given authority in the matter of marriage to the man, so it is not permissible for a woman to arrange a marriage for herself or for anyone else; rather her marriage must be arranged by a wali or guardian. Women have nothing to do with guardianship in marriage; rather that is only for men, and is basically for the father, who takes precedence over others in arranging the marriages of his daughters. 

Imam Abu Bakr al-Qaffaal ash-Shaashi (may Allah have mercy on him) said: 

The reason being – and Allah knows best –that women have a natural inclination towards men and they are not able to judge people well, and that could lead them to choose one who is not an appropriate choice (for marriage),therefore Allah entrusted the woman’s case, with regard to her marriage, to her father. If the matter had been entrusted to her, there would be the fear that she might marry someone who is not compatible with her.

End quote from Mahaasin ash-Sharee‘ah, 247. See also the answer to question no. 2127 

If the father has the right to guardianship in the case of marriage, and the marriage contract cannot be done without him, then it is only logical that his opinion should take precedence in the matter of choosing a husband for his daughter, especially when he in most cases will be better able to judge their character, thus he will be able to find out about the suitor and reach the right conclusion. 

But that does not mean that the mother has no say in choosing a husband for her daughter; rather she should be consulted about the matter and her opinion should be taken into consideration, because perhaps it may be correct and some things may be apparent to her that are not apparent to her husband. 

To sum up: 

The basic principle is that the father’s opinion and choice takes precedence over that of the mother, but that does not mean that the mother’s opinion should be ignored altogether; rather it should be taken into consideration so so that she does not feel left out and to convince her of the father’s point of view, and that he is responsible with regard to their daughter’s marriage. Thus the choice is up to your father, especially if he is known to be of sound reasoning, but he should not make her feel left out; rather he should respect her opinion. 

And Allah knows best.