Letters for Mom. Any ideas on what to write to her?

Oh, my 'out of state' sister used to do this for our Mom. She would make cards from construction paper. And she would tape a photo of the two of them to the inside. Since your Mom has memory issues, her ability to focus is probably compromised. It is more about letting Mom know you love her than the actual message.
September means back to school, do you have a favorite school story? Ditto for Halloween, Thanksgiving, December holidays, vacations, etc. Keep it short. Tell her you love her. A sentence or two about a shared activity -- baking cookies, scouts, sports whatever. Then you can put a picture from a magazine/internet of that thing. A platter of cookies, trick or treaters, a beach, etc.
Keep it simple, it is more the excitement of having something to hold and look over. If the team at her memory care does their job, she will be shown the card/letter over and over --- and it will be new EACH time to her.

You won't be able to send her post cards, or emails that the NH could print off for her? Goodness! I don't like to ask where you're off to (could it be that you could tell me but then you'd have to kill me..?)

Critical responses do not help anyone.
Any associations with her life - pets, hobbies (dancing, cooking, gardening, chess, sewing, horse races (!) ) - cut out pictures in magazines and put in a card and write something about it. Anything to associate with what she loves. Even a I LOVE YOU might be enough. I'd keep the cards simple. Date them (add the day of the week too as so many forget 'what day it is'.

buy a box of cards with photos that she will find pleasing with blank inside. Or make your own or you can buy ones for your printer. Then write a letter about a subject that taps into her long term memory, e.g. family vacations, going out to eat, etc.
Be sure to enclose a photo of yourself or your family. Then she can look at it independently of the card and show it to others more easily than a card.
If she has someone to take her out of her facility, then you could send a small gift card for a restaurant or store she likes to visit.
If it all possible, have a backup visitor scheduled before you go.

I don't think that I would necessarily write a letter. Depending on what her mind is capable of, I think a simple photo "scrapbook" type of thing would be good...a picture of a heart for Valentine's Day, shamrock-St. Patrick's Day, etc. Put a big "I LOVE YOU, MOM" on each. Even draw pictures of animals, whatever you know she likes.

I would suggest buying greeting cards in a package of several cards. Take pictures of family members and keep the letter part shorter. I would creat a template on the computer and add information about what you did that week or what people are doing in the pictures. This way it's a simple system that won't take you hours but make her feel special. Write a special note on the card but make your letter a standard template with information about your day to day life.

Just write to her the things you plan to do while traveling, places you may plan to visit, restaurants that may have been recommended to you. Write several approaching a new topic every day. Your letters need not be long, the idea being that in hearing, or reading what you have written will give her a loving connection to you. Silver Threads

Before Mom was put on a Seroquel generic for anxiety and agitation, I used to leave her a simple note when I went out and a nurse's aide came to stay. The note said I'd gone out, where and when I'd be back. I attached it to the fridge door with a magnet and the aide would show it to Mom whenever she expressed concern.

The aides told me this was effective in calming Mom down. However, as the dementia progressed, nothing worked and we had to resort to medications. I put it off as long as possible but finally gave in when Mom's quality of life obviously was suffering. (Mine too.)

Blessings for an effective solution to your challenge.

Yes, for residents of a NH especially Moms , letters that speak of happy childhood memories, the favorite pies cakes she baked, things you couldn't find and she helped you look for them, trips you took, it goes on and on. Silver Threads

Years ago as a young adult, I wrote letters to both of my grandmothers thanking them for the love and the memories that I had growing up with them, things like bird watching, one taught me to knit and to bake, or the card games we played, the books that they read to me. . . and I heard from family that they were really touched by them, and saved the letters.
So I think If you just thank her for some happy times in your life, she might be able to remember those. My mom can't remember what she ate 5 minutes after eating it, but she does remember a lot of fun stuff from my childhood.