There is nothing wrong with a woman wearing black during her monthly period. This is for two reasons:
The basic principle is that things are permissible and there is nothing in sharee‘ah to say that a woman cannot wear any particular colour during her period, whether that is black or any other colour. So it is essential to adhere to the basic principle.
The fact that the Ka‘bah is covered with black is not a matter that enjoined, because there is nothing in the Sunnah to that effect. Rather it is the custom of people at present. If the Holy Ka‘bah were to be covered in some other appropriate colour, there would be nothing wrong with that. Black is not a special colour for the Holy Ka‘bah such that it would form the basis for a ruling that it is not permissible for a menstruating woman to wear black.
Rather, if we were to assume that the colour of the covering of the Ka‘bah is something enjoined, that would not in itself indicate that it is forbidden for a menstruating woman to wear black or that it is disliked for her to do so.
One should beware of things that are said or spread in rumours stating that something is haraam or halaal, when there is no basis for it in sharee‘ah. Allah, may He be exalted, says (interpretation of the meaning):
“Say (O Muhammad to these polytheists): ‘Tell me, what provision Allaah has sent down to you! And you have made of it lawful and unlawful.’ Say (O Muhammad): ‘Has Allaah permitted you (to do so), or do you invent a lie against Allaah?’
60. And what think those who invent a lie against Allaah, on the Day of Resurrection? [i.e. Do they think that they will be forgiven and excused! Nay, they will have an eternal punishment in the fire of Hell.] Truly, Allaah is full of bounty to mankind, but most of them are ungrateful”
Al-Haafiz Ibn Katheer (may Allah have mercy on him) said:
Allah denounces the one who forbids what Allah has permitted, or permits what Allah has forbidden on the basis of mere opinions or whims and desires, for which there is no evidence. Then He warns them against that on the Day of Resurrection and says (interpretation of the meaning): “And what think those who invent a lie against Allaah, on the Day of Resurrection?” i.e., what do they think will happen to them on the Day they return to Us, the Day of Resurrection?!
End quote. Tafseer Ibn Katheer, 4/276
And Allah knows best.