Could someone please recommend a basin for hair washing, for a bed-bound 85 year old woman.

debster84: A reasonably priced bed pan should work.

Amazon has inflatable options.
nflatable Hair Washing Tray for Sink and Bedside - Comfortable Portable Shampoo Bowl for Spill-Free Rinsing - Wash Basin for Haircuts and Coloring at Home for Elderly, Kids, Bedridden and Disabled

I found a basin that inflates and can be used in the bed. There's a tutorial that comes with the basin. Amazon. I don't remember what is called exactly.

Depending in how thick her hair is, a mildly soapy, water saturated hand towel rub followed by a larger dry towel can do a satisfactory freshen up.

I would just recommend the waterless shampoo and conditioner caps that you can buy on Amazon or walmart.com.
You warm them for a few seconds in the microwave and then you put them on her head and rub/massage it around. and then just take it off and comb out and towel dry her hair.
They work great. I used them on my bedridden late husband when needed and also for myself after having my hip replaced.

A bedpan (about 15.00 on amazon) used only for this purpose works very well. Pad the middle front area of it with a washcloth and gently rest the neck in that area. Use very little low sudsing soap. Rinse with pitcher of warm water. No pillow under head during this procedure, but be certain that the neck is well supported with rolled towel.

Is mom on Hospice? (if not call one, or two and ask them to evaluate her) If so the CNA will do bed bath and will wash hair. That way you do not have to deal with it. The CNA comes 2 to 3 times a week. (It does help if mom's hair is shorter)
If mom's hair is shorter you can use a washcloth and a "no rinse" shampoo. That way if you do not get it all out it will not cause itching or irritation. Place a larger towel or better yet an absorbent under pad around her neck. Use the wash cloth to dampen the hair, put the shampoo on the washcloth and gently massage the scalp then rinse the cloth, or get a clean one then use clear water on the head to remove the shampoo.
And, ask yourself how often mom's hair actually needs to be washed.
As a person ages the skin becomes drier and baths, showers and washing hair can make skin drier so daily baths are not needed. (State of Illinois requires 2 baths or showers a week in a facility. I learned that when my Husband was in rehab)

Hospice offered us a bath shower cap that holds shampoo and you massage the hair while its in. We never tried it because our caregivers use a basin of soapy water and a washcloth to wash her hair and then rinse the soap out with the washcloth. Place a bath towel under her head. My aunt is completely bedridden and unable to lift her head.