Renting an apartment here without a garantia, without you yourself (or depends on who is signing the lease) living there is going to be difficult, especially when we are talking about a mother with her sick child. Even paying in advance. I won't say it's impossible, but it would be a very long shot, I believe.
If you pay a year in advance, the owner of the apartment will have a year paid for, but the owner knows that at the end of that year he stands a possibility of having a mother with her child in the owner's apartment with the possibility of not seeing a dime after that year. The owner's not going to just believe that you will pay the next year, for example, if the woman decides she wants to stay. The owner's not going to believe that you will be able to convince her to leave if she decides to stay. And with the court system here, the owner knows that it'll take years at least to get her out and in the meantime that one year's payment is going to mean anything to the owner.
I'm not saying that's what would happen, mind you. I'm saying what the owner is going to fear.
It might be a little easier to find another sort of place than an apartment. Pensiones come in different sizes and flavors, and they're all over the city, in all parts of town. There are two and a hostel within two blocks of where I live in Recoleta, in a cheta area. It is harder to find a room in a pension with a baby, but not impossible, (don't know how old the child is). No garantia needed, the higher priced ones (say, 4000 pesos) are a room with a private bathroom and maybe a small are to cook. Others start at varyingly low prices and could include a common bathroom and cooking area - and probably dingy at best.
Do a search for "alquiler de pensiones en caba" (or specify the neighborhood instead of "caba") and see if you see anything that might help.
