My 84 year old mother claims that she feels bugs on her head, in her ears and eyes. Does anyone have a remedy?

Maybe your mom really has allergies and is just plain it itchy.my erst itch when my allergies act up .ask the Dr. If she can have Dipahydramine (benadryl)or Lortidine (claritin) ...but itchy and being picky are typical dementia behavior ....GOOD LUCK...

Check with her doctor for a medication list because some of the meds she's on may  be or quite likely are causing these hallucinagenic thoughts.

It could actually be a bug or parasite infestation, so she should see her doctor. But, far more likely, it's hardening of the arteries (atherosclerosis.)
Both of my aged ladies insisted that they "heard" and "felt" what they thought were insects. One was sure the carpeting was infested; the other could "hear" bugs clicking quietly near her head.
Both, unfortunately, had restricted blood flow to the brain. This can often alter vision, hearing, and odd skin sensations. One passed away of stroke at 83; the other of the same cause at 94.
I'd have her see her doctor soon, to narrow the diagnosis and bring her relief. Even if the "bugs" aren't real, they're real enough to drive her crazy.
Doc may prescribe warfarin and/or other blood thinners as well as a statin; or other meds if she does have botfly or other parasite of the scalp.

Have her checked her for a urinary infection; check her meds, has she started any news one? She may be getting hallucinates from either source. At her age it can happen.

Is there a possibility that a sensitivity has developed to a soap or shampoo, and that dry skin has become a problem?

Feeling bugs happens fairly often. There are medications that can help with the creepy crawly feelings, so let her doctor know what is going on. It usually comes from nerves that are sending faulty signals to the brain. The medications help to soothe the nerves.

If it didn’t start out as a stress issue, it probably became one quickly for her. Have her medications recently changed? Feeling or seeing bugs can be caused by the addition of meds or withdrawal from them. Pain meds are notorious for causing these delusions. No matter how much you tell her there are no bugs, she won’t believe you. Call her PCP and ask them if there is an anti- anxiety med she could take.