Does anyone have a suggestion based on experience? Or thoughts on what DOESN’T work well?
I got my daughter a mesh one from Amazon. She said it worked well.
I bought a very simple white plastic caddy at BBB for S17. It has a handle and two divided compartments. Both sides have holes at the bottom for drainage. S17 just came home with after two years and put it right into our bathroom. It holds his toothbrush, toothpaste, hairbrush, razor and supplies, loofah, soap, shampoo and a couple of other personal toiletries. He is happy with it and it cost less than $15 and has survived 2 school years.
We bought a bathroom caddy at BBB for DS. He never used it…at all. So…we hung onto it for DD. When it came time for her, she declined it as well…so we donated it to Goodwill.
My kids said…it wasn’t hard to carry a towel, body wash, and shampoo without anything to put it in. It wasn’t hard to carry a toothbrush or shaver either.
When I was in college back in the Stone Age, we had “potty pails” which were nothing more than a small beach pail. You can get one of those at the Dollar Store…for a dollar. Or one of those plastic totes that looks like woven plastic…usually hold beach toys. That would too.
And unrelated…if it’s a community shower don’t forget shower flip flops. 
If your college student will wear them.
We sent my son off with a plastic caddy from Walmart and two pair of $1 flip fops for the GAP. They served him well. No need to overthink this one.
Per my older cousin’s recommendation, 30+ years ago I used a pail. It worked pretty well.
My sons both just used their hands. “Guys don’t use that much stuff, Mom”
Back in the day, i just used my hands. I can’t recall ever using anything. Our s used a BB&B plastic basket caddy with a handle. D had a suite bathroom & just left things in the bathroom.