If a daughter chooses her divorced mother after reaching the age of discretion, is her mother then entitled to remarry, or is her father entitled to take her from her mother? And what is the ruling of Islamic law on this matter?

If your waiting period (iddah) after divorce has ended, you are permitted to marry. Your right to custody of your daughter, according to the majority of scholars, is forfeited if you marry a man who is a stranger to her, based on the saying of the Messenger of Allah, peace and blessings be upon him: "You have more right to him, as long as you do not marry." However, if you marry a close relative of the girl, such as her paternal uncle, your custody is not forfeited. If your right to custody of the daughter is forfeited, her custody transfers to your mother, unless there is an impediment, or to the father according to the Hanbalis.