Praise be to Allah.
With regard to things that may be used for halaal or haraam purposes, it is not permissible to sell them to people when it is known or it is thought most likely that they will use them for haraam purposes. If we do not know whether they will use them for halaal or haraam purposes, then there is nothing wrong with selling them to them, because the basic principle is that selling is permissible and is not forbidden. This has been discussed in the answer to question no. 75007.
Renting a shop to someone who sells permissible goods in it, such as men’s perfumes for example, and other goods that may be used for halaal or haraam purposes, such as cosmetics, depends on what is usually the case. If most of the products are of the second type, and most of the people in that country use them for haraam purposes, then it is not allowed to rent the shop to a person to whom this applies, so that you will not be helping in something haraam, because Allaah says (interpretation of the meaning):
“Help you one another in Al‑Birr and At‑Taqwa (virtue, righteousness and piety); but do not help one another in sin and transgression. And fear Allaah. Verily, Allaah is Severe in punishment”
But if what usually happens is sale of permissible goods in the permissible manner, then it is permissible to rent the shop, based on the usual situation, and so as to ward off hardship and difficulty, because most sellers cannot avoid selling things that may be used for halaal or haraam purposes.
There is nothing wrong with selling perfumes containing alcohol, because one cannot be certain that they contain intoxicating alcohol, or that it remains intoxicating.
See the answer to question no. 1365.
You should treat your mother kindly and strive to call her to pray, and teach here what she does not know of the rulings of Islam.
If you think that speaking to her about this shop will put her off, then do not speak to her about it; let your speaking to her be about things that are more important, such as praying regularly, in a kind and gentle manner, and with beautiful exhortation.
There is nothing wrong with you benefitting from your parents’ wealth, even if renting the shop is not permissible according to the details mentioned above – because that which is haraam because of the manner in which it is acquired is haraam only for the one who acquires it, but in your case the wealth is coming to you in a permissible manner, which is as a gift from your mother.
But strive to advise her and guide her to what is good, because she is the most deserving of people of your advice and calling (da’wah).
May Allaah help us and you to do that which He loves and which pleases Him.
And Allaah knows best.