GORILLA GRIP Original Premium Anti-Fatigue Comfort Mat, Phthalate Free, Ships Flat, Ergonomically Engineered, Extra Support and Thick, Kitchen and Office Standing Desk, 32x20, Black https://www.amazon.com/dp/B01B868D1M/ref=cm_sw_r_cp_api_glc_fabc_e2Q1Fb9E6V761
Use it every day. Really like it.
I had the same problem with that type of interlocking fitness mat on a slick floor.
I ended up trying 4-5 different types of mats at home, and this was by far the best in terms of safe size, comfort, and not skidding.
https://www.amazon.com/gp/product/B01B868D1M/ref=oh_aui_detailpage_o03_s01?ie=UTF8&psc=1
Will your new laundry room have a sink? That's the #1 thing I wish I had in my basement laundry area (and will probably add at some point, but it's a bit tricky as our basement is below grade and the sink will need to have a pump to move the draining water to where it needs to go).
As /u/LadyAlexTheDeviant said, a big flat space for folding laundry is also nice to have, at a good height for when you're standing. Put an anti-fatigue mat where you'll be standing to fold and/or iron, to save your back and feet. We have several of these and they're so comfy, and easy to hose down if they get dirty.
For hanging laundry inside, folding racks are nice because you can have tons of space with a few racks, and then you can easily store them away when not in use. A ceiling-mount one sounds awesome, but it can only hold so much.
Do you have outdoor space you can use for drying laundry when it's nice out? That's faster than indoor drying (and, in my opinion, there's just nothing like the smell of fresh sun-dried laundry!). I have one of these, which is handy because it can be popped out of the ground and stored when not in use.
https://www.amazon.com/Original-Anti-Fatigue-Phthalate-Ergonomically-Engineered/dp/B01B868D1M
I have no affiliation with this product, but my feet and back highly promote it.
For context, I work from home at a Jarvis desk, so I'm standing roughly half the time I'm working and more if I'm playing tunes that day. This made a huge difference.