Mom still desires to live on her own. Any advice?

How long has she been with you, now?

The request to "go home" is common. It is not to be taken literally and you can't teach her what reality means now.
She wants to go back in time to a place where she was young, probably surrounded by people she loved, and was happy. This place is no longer real or possible.

All you can do is speak calmly to her and get her to talk about that time/place and describe it to you if she can. You can tell her that if you could, you would make things like they used to be. But mom, you're here now, we love you, and everything will be OK. Then have a diversion at the ready.

Be prepared to do this A LOT. Remain calm, it's not intended to annoy you, but you have to see this kind of thing for what it is and don't stress yourself over it. It's just a comment from mom to express something far more complicated than "take me home".

There are a lot of articles on this site about this topic if you search for them in the search box. Keep checking back!

As mentioned above, this may be sundowning. Is she on any meds for anxiety ir depression? Does she start saying this at a certain time of day
? Sometimes a distraction that starts before she feels this way (a DVD of a favorite tv show, recorded music, a walk) helps

When my mother-in-law started says that all the time, I found out all she needed was to be validated as to WISHING she could go back home again. I started agreeing with her and saying I WISHED she could go home again. I WISHED I could turn back the clock and make it happen for her, but I can't. She also has dementia, so it was a constant repeating of the same thing until I thought I'd lose my mind. But she IS losing her mind, so I bucked up and kept repeating that same thing until she stopped lamenting about moving back into her house. She has 'sun-downers' so anytime after 1pm(ish) she's back in the same loop of wanting to go back home. Can't be helped. Sorry, and good luck.