Need a lodging place for my 87yr old mom for three weeks while she waits on an apartment to become available. Any suggestions?

I would make contact with some Airbnb owners, some will reduce the price for longer stays. Another option is an extended stay hotel, sometimes they offer a free day after so many.

A friend had a friend who sold her house to pay for a senior living facility but the move-in date was a month away. So my friend rented this lady a room in her house for a month. You might know someone who could use a little extra cash and would take your mom in.

The only other possibility I can think of is a short term AirB&B furnished rental in her area. Either an affordable studio or a room in someone's apartment - try airb&b or vrbo.
Good luck.

Technical I bet she is “homeless”, so if there are shelters or temporary placement facilities, she might be able to stay there.

Where is she currently living, Butterfly, and why cannot she stay there? I mean even if they are evicting her it could take that long, or with perhaps just a week of a motel to worry about. I agree with not taking her in your home. And I think she shouldn't go into anyone else's either. So you are down to respite center for a month or a motel. Some will give a monthly rate. You are correct, expensive. Consider a go-fund-me.

Have you looked into long term stay hotel rooms? More like a studio apartment than anything else. DH traveled a lot and often his company kept one or two of these suites on hold for the people who would be out of town for weeks on end.

Much less expensive than a regular hotel, but still pricey. It's also entirely possible that during the holidays these types of places are not easy to find. IDK. Worth a call, I guess.

Based on how independent she is, can she live alone and manage the daily stuff of living, or would you be going to take care of her day after day?