Mom has extremely dry skin no matter what moisturizers I use. Can anyone recommend something that works?

Part of the problem is the dry skin, and part is the urge or habit to scratch. Good advice here about dry skin. For the irritation and the habit or urge to scratch, you could try some of the anti-histamine creams or pills. There is a pill, here called Restavit, containing Doxylamine Succinate 25 mg. It is used as a short term sleeping pill, and works for people who are made drowsy by anti-histamines (as in ‘do not operate heavy machinery’). It might be useful for a few nights, which could help to break an itching/ scratching habit.

If she can tolerate, I recommend a coffee scrub (coffee grounds mixed with coconut oil) or a Japanese scrubbing towel to exfoliate. Obviously ONLY if her skin can handle it.

Then I'd follow up with argan oil (all over), the Aveeno oatmeal lotion (all over), and the "miracle hand repair" lotion on arms and legs.

Be careful because applying that to bottoms of feet can make walking tricky, especially if on the hands and they need to grab bars or canes or walkers.

Honestly, I found a very good product from ScentsationalJourneys her name is Sarah Torrance. She has a salve that is fabulous. Her aging mom uses it and loves it.

ingest some vitamin E? benedryl?

How is the air in your home? I good humidifier might help. Be careful of the soap she uses. Also pay attention to the foods she eats. If nothing works look it up on line, in the search bar. Tons of answers there.

Lachydrin lotion or any generic equivalent lotion with lactic acid

I second cerave. Used to be rx only. Sarna for itching.

try “CeraVe Diabetics Dry Skin Relief.” It’s on Amazon. It’s good.

I can also attest to aquaphor. The duct tape of dermatology! =)
Also... Avon's moisture therapy line. The calming relief is for itchy dry skin. Also, their intensive line is exceptionally good. The hand cream, lotion and extra strength cream.
Now. The other thing is to get her skin moisturized the second she is mostly dry from the bath/shower. Even using baby oil can help.
Also, (I'm not the natural everything organic type.) using a gentler soap /body wash could help. The itching could be from preservatives in shampoo /conditioners and body wash. It is very common for an allergic reaction to certain preservatives.
Another thing it could be is a reaction to the chlorine /chloramines in the water. There are multiple varieties of shower head filters. Sprite I think is the name of one. Reasonably priced. And change the filters every three months or so.
You can also try a gentler laundry soap. All free and clear is one to try.
One more thing...Lowering the temperature of the bath even just a little bit can help too.
I hope this helps!
Sparkles

Aveeno has several products for dry shin and itching. There is a shower wash soap too.
The brands mentioned above work well too.

Cetaphil and Aquaphor both are proven, non-allergenic and fragrance free.. When she does have a cut or scratch, generously slather on bacitracin.

Keep her nails manicured and filed smooth so they don’t snag her skin as easily and give her things to do with her hands so she doesn’t scratch out of boredom.