Info about a stray cat?

My moms friend invited my mom and I over. We were sitting on her back porch when a stray kitten is to come up on the deck. It came over to me, and rubbed against my leg. I pet her head and she laid down and stayed for a while. She was very skinny, so I don't think she belongs to anyone. her claws are long and greenish, witch is probably from plants and grass. I am trying to convince my moms friend to take the cat in. She is concerned about her daughter not being able to come over (her daughter is allergic to cats). This poor cat seems like it can't survive on its own! I would not feel comfortable giving her to a stranger. I want my moms friend to keep it, or better yet, myself! Is there any ideas on how to convince my mom? we already have 1 cat, so there are no allergy issues. I offered to pay for her vet trips/food, but she still won't budge. Even if the cat (smoky is what we call her because of her grey pattern that looks like smoke) can survive on its own, it deserves a nice, loving, and warm home. how can i convince my mom to take her in!?!