What should a woman do if menstruation begins before she performs the Witr prayer of the night prayer?

Witr prayer is a Sunnah (definition), not an obligation. There is no sin upon a woman who misses it or misses making it up. If a woman sees menstrual blood after the 'Isha prayer and before the Witr prayer, she should not pray Witr. If she becomes pure, it is recommended for her to make up this Witr prayer, based on the Prophet's (peace be upon him) saying: "Whoever sleeps through his Witr prayer or forgets it, let him pray it when he remembers it." The preponderant opinion among us is that whoever misses Witr prayer should always make it up, and should make it up as an even number of rak'ahs. So, if he used to pray Witr with one rak'ah, he should pray two rak'ahs. If he used to pray Witr with three rak'ahs, he should pray four, and so on. This is based on the hadith (definition) of 'Aisha, who said that the Prophet (peace be upon him) "If he missed his portion [of night prayer] at night, he would pray twelve rak'ahs during the day."