My mom's most repeated thing she says is how lonely she is. Any suggestions?

Much of the stuff our elders with some dementia say is more or less rote habitual speech.
In fact this is often true of ALL of us, whether we suffer from any brain disability or not. Notice who comes to work always sighing and down, and who shows up always chipper and perky. You will eventually conclude that it really has little to do with their real lives.
We develop habits. It is often little other than that.

I don't know if it is true of your Mom or not, but it is said that we die much as we lived. Our last years are either ones of depression, resignation, deep sighs and looking at the downside (and there's plenty of it to see, I assure you at age 80)
OR
we are more or less content, enjoying those things we enjoy while acknowledging that we do so with a few more aches and pains.

I think if you are too "tortured" by this (using your word) you may want to seek a few hours of counseling with a licensed social worker in private counseling practice. They are great at life transitions work.
We cannot change people, but we can change our reactions to them.
You didn't cause any of this. You can't fix it.
And honestly you are already doing more than any three people put together. Doing twice what you are doing is very unlikely to change a thing.
Not everything in life can be made all happy happy, so just take her hand and say "I am so sorry. I wish you felt a bit better". Listen to her! Don't try to change things for her. It is her reality and she should be given the dignity of it.

Throughout our lives we choose whether to look more at the light side or the dark. There's plenty of EACH to see.
I surely do wish you good luck.
Remember, you don't want to mirror what your Mom is doing. You have a choice which side to concentrate your energies on.

"How do I let it go when she says she is lonely all the time?"

You immediately translate her "I'm so lonely" to "I forgot that I saw so and so 20 minutes/an hour ago/yesterday" because that is exactly what happened.

That's how you let it go.

If shes lonely a facility is the perfect place for her. Social interaction would be beneficial for her. Start touring places for her to move to .

If mom said she was living on the sun and it was too hot, she needed to go home, would you internalize such a thing and allow it to torture you? Or would you chalk it off to brain disease, knowing she's now existing in another reality which you have NO control over to change or to "fix"? My mother insisted her girls were moving her to a new hotel every night from her suite in memory care. And taking her to parties every night at different entertainment places. She was tired of all the hubbub and just wanted to rest. What was I supposed to do about it? Nothing. Just listen, agree, nod my head, change the subject, give her a snack or an extra Ativan, and wish her a good evening.

We all lose when dementia comes into play. But if you are going to buy into her demented delusions, you may wind up suffering more than she is in the end. You're doing everything humanly possible for the woman, so let it all go and allow God to help you accept mom's situation now. She may perceive herself lonely 2 min after a visitor leaves, which doesn't make it a reality. Know that you are a wonderful child to her, and doing your level best in a difficult situation.

I suggest you read this 33 of booklet online about managing dementia and what to expect with an elder who's been diagnosed with it.

Understanding the Dementia Experience, by Jennifer Ghent-Fuller 
https://www.smashwords.com/books/view/210580

Jennifer is a nurse who worked for many years as an educator and counsellor for people with dementia and their families, as well as others in caring roles. She addresses the emotional and grief issues in the contexts in which they arise for families living with dementia. The reviews for her books are phenomenal b/c they are written in plain English & very easy to read/understand. Her writings have been VERY helpful for me.

The full copy of her book is available here:

https://www.amazon.com/Thoughtful-Dementia-Care-Understanding-Experience/dp/B09WN439CC/ref=sr_1_2?crid=2E7WWE9X5UFXR&keywords=jennifer+ghent+fuller+books&qid=1657468364&sprefix=jennifer+ghent%2Caps%2C631&sr=8-2

Good luck!

Think about what “lonely” means. People are often lonely in a crowd. Lonely is when you don’t feel anyone “inside”, anyone who knows and understands what you are going through. With short term memory loss it is almost impossible to hold on to that person who knows, loves and understands you. May I suggest then, the Miracle of music from her youth? It does wonders for mood.

My grandmother was a very socially active woman and fortunately never suffered from any level of dementia but she often mentioned feeling lonely, I don't think she ever got over the loss of her dearly loved husband. Being surrounded by other people does not necessarily cancel out the deep loneliness that comes from the loss of a spouse, siblings, friends and other close relationships, at best it distracts temporarily. I think all you can do is commiserate, maybe occasionally ask who she is missing and take a walk down memory lane with her.

Does she have an adult daycare center in her city or near? If so, you can have someone take her there up to 5 days a week, and they can stay up to 8 hours per day. They serve breakfast, a snack and lunch and always have great activities to keep the folks busy. The people that work there are great and take great care of their clientele. And if money is an issue they have programs available to help with that as well.
The one in my city even has a spa like room where they will bathe and do the persons hair.
It would be great opportunity for your mom to be around other people and have some fun. Most people leave quite tired from all the activity. They're worth every penny.

Never, ever thought I would move my mom to memory care! She loves it, is happier and smiling every day. Haven't had one bad or sad or mad moment on the phone or at a visit. She was actually in an independent living facility and got covid. No physical issues but it did a number on her mind. She started "escaping". Moved her to my home that she has always loved and she "hated" it and I was keeping her "prisoner" so she started walking away several times a day. She was either crying or mad as a hornet and wouldn't even look at me. So my brother and sister said, "enough". After a month of mourning because I wasn't seeing her everyday, I realize it was the best for everyone. I see her once a month and talk to her twice a week, call the facility every other day to make sure she is happy and it works for us and her. Make the decision to do what is best for her and then move on. There is no perfect solution in the world of alzheimers.

We used a huge wall calendar for my mom when she was in rehab - always forgetting all the people who showed up to visit!! We’d write our names on the days we came and when she’d start with “where have you been?” we just had to point to the calendar and she’d stop. We also used a write on/wipe off sheet (they make big rolls of this stuff that easily stick on the wall) and we wrote every visitors name on it … here’s your visiting team! We may have had to remind her to look at them but it worked overall!

My husband's aunt, 94, says the same thing ALL the time and even cries she is so upset. She has hired caregivers whom she loves plus family regularly visiting each week. She remembers absolutely nothing. The only thing you can do is remind her who visited when you see her. We now buy flowers every week & she seems to connect that with visitors.