Hiring a friend's sibling to stay with elderly loved one, what are the legalities involved?

Aside from the legality/green card issue, if this person doesn't have a good command of English, I'd pass. Also, as I've been an employer for decades, our policy was to never hire friends or relatives because it makes it tricky to fire them.

"your employer cannot demand that you show a green card, if you have other documents listed on the I-9 Form showing that you are authorized to work in the U.S."
 
Its illegal to hire illegals. So you need proof that a person can work in the US.

Will there be a language barrier? I may think twice about hiring this person. Yes, u would be responsible for all payroll deductions and seeing it gets to the right agency every quarter.

The easiest way to do this is to use a payroll agency.

They will do all the verification and reporting to ensure that your employee has all the required documents to work in the USA.

It will cost you whatever hourly wage, you set the rate, plus around 25% of that but, this includes all of the taxes, insurance and filings required. Often they can offer the employee health insurance as well.

You want to check with your homeowners insurance to find out if you need an umbrella or a rider to ensure you are protected on that end. Be sure and explain how this person is hired and paid because having workman's comp insurance is a deal changer for your required insurance, a good deal changer, the employee can not sue unless they can prove negligence when workers comp is on board, super important that is.

That's how I recommend a private caregiver is paid, it simplifies and protects you as an employer and offers added benefits to the employee.

In California, while you can access computer information when, say, hiring a farm worker, you cannot legally question people about their immigrant status.
As you are concerned with this, I would ask an elder law attorney for an opinion, paying for an hour of time you can ask any other questions you have thought of regarding care, also.
Do know, that even illegals often pay taxes. Go figure. It is often withdrawn automatically out of any paychecks they receive.

So your mother would pay and report payment just as she would for anyone else.
The rest of the issue is up to this OTHER family.

Think about it. Do you question the legal status of anyone ELSE you hire? Unlikely. So just go on about your business, and if this woman is good at her job, pay her the wages that are minimally dictated in your state--a fair wage. Give her the tax forms and report. Voila! Your job is done.

Her status to my mind is her business, not yours. Just my humble opinion.