Where do you live? Depending on where I would recommend insulating and heating or clothing and shoes wont dry and will be super cold.

Thanks. We are in Seattle.

No HVAC anywhere near. I was thinking of radiant floors, but that would require a new circuit from the breaker box. I'm not sure I am comfortable to DIY and it would probably double the budget for a tiny closet. We will probably have a small space heater for heat if necessary.

We live in Vancouver and our garage isn’t heated nor insulated. Coats don’t dry and neither do shoes.

That is my biggest worry- that we finally get the coats and shoes out of the kitchen only to put them in a wet refrigerator. Our house is very old, so I'm not sure the exterior wall is even insulated, which could help!

This new room will have insulation. May try to put a boot dryer or space heater in there to run when we are in it.

What about brick?

" that we finally get the coats and shoes out of the kitchen only to put them in a wet refrigerator. "

That is exactly what you will be doing.

My mud room is in the basement, and there is no heat duct. I am on a farm, and get wet and muddy. I am going out in weather appropriate gear, so if it’s raining, I am in a raincoat and boots. They drip dry just fine in a few hours. The only thing that gets wet that I need to throw in the dryer are my Carhartts bibs; they are warm but not water resistant. Oh and the dog. He’s an Aussie and takes a few hours to dry when he gets wet. Your set up will be fine, just keep a laundry basket in there for the few times you will need to toss a couple things in the dryer. I have ceramic tile and it mops up great. A couple scatter rugs allow me to remove boots (into a boot tray) and get into slippers without standing on the cold floor. It’s really not an issue at all. My rain gear drips dry plenty fast enough. It’s honestly better to dress and undress for below freezing weather in a cool room rather than the heated house.

So I would go with large ceramic tile in a color close to the dirt you will be bringing in. Doormats or rugs that are easy to clean. Take some pictures and post, please.

If you live in a wet cool climate, an unheated room will also be wet and cool unless you do something to change it. I think I’d be reluctant to take off my coat and boots in a cool and damp room; I’d just do it in the kitchen like always.

What about a wall-mounted hardwired heater of some type? One with a thermostat. Google it. You wouldn’t need a very big one, just need to take the chill off.

Just use a 120 volt wall "bathroom" forced air heater. It will give you an option to heat up that small space to get things to dry and it would not require a new circuit. https://www.homedepot.com/p/Cadet-120-volt-1-000-watt-RBF-Series-In-wall-Fan-forced-Electric-Bathroom-Heater-in-Chrome-RBF101/202696379

As for flooring, a darker natural looking porcelain brick look tile with some variation to hide dirt would be my choice. https://www.homedepot.com/p/MSI-Capella-Red-2-25-in-x-10-in-Matte-Porcelain-Stone-Look-Floor-and-Wall-Tile-5-15-sq-ft-Case-NCAPREDBRI2X10/312146729

Definitely need it insulated and heated. Trust me. Don’t make the mistake we did in the garage. Now we do have a mud room and a huge closet so it’s fine and it’s a great winter storage area for drinks and food but I wanted it to be a place for wet stuff to dry and it 100% is not that. Seattle and Vancouver have very similar climates

I agree do some serious thinking before doing this .BTW I live in Creston BC not really damp but can be a bit and often our bathing suits do not drip dry overnight in the summer I can't imagine trying to do that in the winter in an unheated place even in Seattle