All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
After completing a marriage contract, the wife has the right to give up her Mahr (bridal money). Canceling the Mahr prior to the marriage contract is not correct since the Mahr is among the conditions of a valid marriage. Giving up the Mahr before the marriage contract is nonsense, how can she give up her Mahr before owning it?
Ibn Qudaamah wrote in his book Al-Mughni: “It is permitted for a woman to give up her total Mahr or part of it, or present it to her husband. There is no disagreement reported in this matter. Allah Says (what means):
• {… unless they forego the right or the one in whose hand is the marriage contract foregoes it.} [Quran 2:237]
• {But if they give up willingly to you anything of it, then take it in satisfaction and ease.} [Quran 4:4]
What the questioner mentioned that it was among the peculiarities of the Prophet to marry without Mahr is correct. Imam Ibn Al-‘Arabi mentioned when interpreting the verse: {…and a believing woman if she gives herself to the Prophet [and] if the Prophet wishes to marry her, [this is] only for you, excluding the [other] believers.} [Quran 33:50], that Allah made lawful for his Prophet to marry a woman without giving her Mahr. It was yet another favor given to the Prophet upon other members of his Ummah. No one shares that special favor with him because marrying a woman and giving her Mahr is lawful for any individual of this Ummah.”
The author of al-Gurar Al-Bahiyyah wrote in his book while mentioning the characteristics of the Prophet : “And he was given the right to marry a woman without giving her Mahr as happened with the woman who offered herself in marriage to him as mentioned in the verse.”
As for the name of the woman who presented herself to the Prophet there are more than one name that are mentioned by the scholars. Some are of the opinion that she was Maymoonah, others believe that she was Umm Shareek. It is also said that she was an Ansari woman.
Allah knows best.