Firstly, she would be eligible for Child Benefit if she meets the following conditions that is applicable to temporary resident like your friend.
a temporary resident (as defined in the Immigration and Refugee Protection
Act) who has lived in Canada throughout the previous 18 months, and
who has a valid permit in the 19th month other than one that states “does
not confer status” or “does not confer temporary resident status.” If this
is your situation, do not apply before the 19th month;
AND
To get the CCB, you have to file your return every year, even if you did not have income in the year. If you have a spouse or common-law partner, they also have to file a return every year.
Based on her circumstance that you explained above, I think she would be eligible to apply for Child Benefit AS LONG AS she has filed income tax returns.
Secondly, her family member in Vancouver will have to apply for LMIA in order to sponsor her as a live-in caretaker. The process will take a couple of months (give or take two to four months). Once they have received a positive LMIA then they can sponsor her a live-in caretaker. But LMIA application process isn't free as the employer which is her family member in Vancouver in this case will have to pay $1000 to the government as an application fee and will have to hire either immigration consultant (can be $2000-$3000) or lawyer (lot more expensive than consultant) who is familiar in the process. If they are OK with all those fees then they can go ahead and make an application but note that there won't be refunds should they receive a negative LMIA...
Another option she has is, go to Vancouver and try to find someone who is looking for live-in caretaker. Honestly, I'm not sure whether she would have to apply for another work permit if she does find someone who has a positive LMIA and willing to hire her as her work permit is closed work permit (meaning it is tied to a specific employer which is her former employer who terminated the contract with her). It would be better for her to find a good consultant and see what other options are available to her.
The best solution I can think of is, find someone who has a positive LMIA and willing to hire her, and get two years of working experience as a caretaker, and after two years apply for PR.
Though she has family members in Canada, they can't really apply for family sponsorship. Also, humanitarian and compassionate ground appeal is out of option for her though her child is Canadian. Immigration officer can tell her to give her child to her family member in Canada and send her back to her home country unless she wants to bring her child with her.
Wish the best.
