I played with adding rugs to the room shown on Amazon, found the ones I liked generally had a light background and a subtle geometric pattern. (But that's strongly influenced by the warmer colors, creams and golden ambers, in that room.) Here are a few of the better ones that might work with your cooler color scheme:

These are all so subtle that you could probably just get carpeting bound and get a similar look. Online offers lots of choices, but it's easier to judge color in person.

Best wishes as you make a home for your mom. This is not an easy road to walk, I know from experience.

I will echo the concern about having an area rug. It adds to the chance of a fall. And if she needs a walker, it will be worse.

You may want to have a government agency (if available) come in to access the home for safety and provide advice (if you haven’t already). When my mother was still in her home, they made all kinds of safety/convenience recommendations we hadn’t thought of.

When she moved to a seniors residence no rugs were allowed. As my mom‘s dementia got worse, she would fall getting out of bed. If it helps down the road, we were allowed to add a large spongy mat, which we tapped down on all sides as well.

https://seniorsafetyadvice.com/are-throw-rugs-dangerous/

Although I haven’t worked with the elderly in a long time I also have to suggest no area rugs. With your MIL’s diagnosis this will prove more problematic over time. If you all felt that you have to have a covering there are different types of carpeting that could work although install would damage the floors. Good luck!

Here are some patterned options that could work for the room.

Thank you all so much for the helpful comments & suggestions. My MIL very much wants a rug in the room & I thought it would be ok while she’s still healthy physically, but we’ll talk to her again & explain the safety concerns. I was avoiding carpeting because I fear she may need a walker in the future. Thanks again!

I'm glad to hear she's okay.

Get her an area rug. Then you can simply roll it up if she has mobility issues later. Beverly gave you several beautiful ideas.