Praise be to Allaah.
Know that the rights of the mother over her child include his being kind to her and treating her well. Allaah says (interpretation of the meanings):
“And that you be dutiful to your parents” [al-Israa’ 17:23]
“And We have enjoined on man to be good and dutiful to his parents” [al-‘Ankaboot 29:8]
“give thanks to Me and to your parents” [Luqmaan 31:14]
Abu ‘Amr al-Shaybaani said: “I asked the Messenger of Allaah (peace and blessings of Allaah be upon him) which deed was most beloved to Allaah. He said: ‘Prayer performed at the right time.’” He asked, “Then what?” He said, “Honouring one's parents.” He said, “Then what?” He said, “Jihaad for the sake of Allaah.” (al-Bukhaari, 5970)
Ibn Hajar said in Al-Fath (10/401): “The Aayah implies that one should honour and obey one's parents even if they are Kaafirs. The Prophet (peace and blessings of Allaah be upon him) explained with regard to another hadeeth that it is good to honour one's parents even after they have died. It was reported that ‘Abd-Allaah ibn ‘Umar met a Bedouin man on the way to Makkah. ‘Abd-Allaah greeted him with salaam, mounted him on the donkey he was riding, and gave him a turban that he had been wearing on his head. Ibn Deenaar said, we said to him, May Allaah bring you back to your senses! They are only Bedouin and they would be content with little. ‘Abd-Allaah said: this man’s father used to be a friend of ‘Umar ibn al-Khattaab, and I heard the Messenger of Allaah (peace and blessings of Allaah be upon him) say that the best of good deeds is for a son to uphold ties with his father’s friends. (Saheeh Muslim, 4629). Al-Ubayy said in his commentary on Muslim (8/496): this means that the best good deed is to give preference to one's father’s relatives over others.
As for the son’s rights over his mother, these include a good upbringing, kind treatment and compassion, which involves fully breastfeeding the child, educating him and spending on him if the father is not present.
With regard to your mother’s rejection of the girl you wanted to marry, you have to find out the reasons why your mother did not like her, because a person’s judgement is a reflection of the impressions he or she has. Perhaps your mother looks at the matter from a different angle than you do. Perhaps she rejected this girl for a reason that is quite legitimate from an Islamic point of view, such as the girl not being committed to Islam. Maybe after researching the matter and consulting with others, she found out that this girl was not the right one for you; in this case you should obey her.
But if your mother rejected her on the basis of her own whims, and not for any reason that has to do with Sharee’ah, or she rejected her because she (the girl) is strongly committed to Islam, then in this case you do not have to obey your mother, but you should try to win her over gently. You could get some good people whom she respects to intervene in the matter if you wish. And Allaah is the source of strength.