My mother has Alzheimer's, can she still sign a health care proxy?

If she is not competent to sign a legal document, there is still hope:
Please talk with a Social Worker, or a Legal help, for instance, available at Area Agency on Aging.
When someone is mentally unable, it is a little more complicated, but, can be done.
It first needs a Doc to sign that the person is not competent, and then, a judge appoints someone to be her POA.
It can be a fairly quick process, if needs be, depending on how good the documentation is about the incompetency, and IF there is no contest as to who to make the POA. .

Does she still understand the concept of appointing someone to make health care decisions on her behalf? I think that is the critical factor ... not the diagnosis.

Is there anyone likely to contest this? Down the road, if the proxy is making a decision is there a relative who might come forward and say, "You can't do that! Auntie was not competent when she signed the papers!"

Does a health care proxy mean that she designates someone to make her immediate health care decisions for her, or does it mean she designates someone to make decisions if she becomes unable to do so later on?

I think her ability to designate someone would depend on her understanding what she is signing.and this depends how fa r advanced the AD is.