Since my father passed away in June 2021, my mother refuses to leave her house for even an overnight stay with one of us. How can we help?

She lost half her self just six months ago. I'll wager she and your dad were married more than half their lives, so it isn't just something you move on from in the blink of an eye.

As long as she's making a little progress here and there, her grieving is not unhealthy and there's no good reason to make her come to your house to make it easier on you. She wants to be where things are familiar, where she feels your dad close be, and where she's comfortable. She may not want anyone to hear her cry at night.

Just be sure she gets to her doctor, grocery store, and has people to talk to for now. Anxiety might be very dangerous for her health now that your dad is gone. Perhaps to continue talking about why she doesn't want to be away from her house would offer insights for ways you can help her. For instance, to get to your house would she have to drive? Maybe she doesn't feel safe on her own to drive. She's lost her companion. It's really difficult making this kind of life adjustment.

It would be better for her to agree to come to your house so she could see everyone. Is this something she did in the past? If it's a new thing, it could be hard for an 84 yo to think about doing that. I'd keep asking and trying to gently tease out of her why she is reluctant. Maybe there are things you can help her with? When my mom goes to stay with my sister for a few days I have to end up helping her pack. She stares at her stuff and can't decide if she has enough, not enough, etc etc. She's always had a hard time with decisions and it's just that much worse now.

Do you think your mom could be a bit depressed from her husband's death? After my dad's passing, mom's been on an anti-depressant. Hard to tell how much it helps with her memory issues, but it's something to consider.

If she getting out during the day? Do your brother's check on her and visit?

I would hope that your brothers are doing their share. I am the oldest and a girl who stayed where I grew up 5 min away from parents. So it all fell on me. Your brothers are closer, so they should be carrying most of the load. Just how it is. No, two hours isn't far away but you also aren't a skip and a jump away. Is Mom complaining she does not see your family enough? If not, don't worry about it for now. If she is, then say that your children cannot just pick up and visit with work and children. Best thing would be her spending a weekend with you.

Its only been 6 months and she is 85. She cared for him and now she has no idea what to do with herself and grieving her loss. She wants her own bed.

2 hours is not that far away for you to continue to visit your mom if that is her preference right now. She may feel closer to your father when she's in her own home, the home they shared, and isn't ready yet to give that up even for a few days.
She is still very early in her grieving, and everyone deals with grief their own way, so for now, just cut your mom some slack and continue to go see her at her home. You can of course continue to make the offer to come stay with you for a while, and hopefully in time, she will. But remember too, that the holidays are the hardest part of the year to get through when you've lost someone dear to you, so be patient and be kind to not only her, but to yourself as well.
I'm sorry for your loss.