My father gave POA to my mother, who is now experiencing demenita. Can he name me (his daughter) at this time? He is lucid.

Since your dad is lucid and aware that his wife is declining, then he can withdraw her from being the POA and give it to you. In one of those not so demented times, this needs to be shared with and explained to your mom before he just up and does it.

Exactly how that is done depends on how the POA is written - you may be able to figure it out yourself, or you may need a lawyer. But most of the itme a POA can resign form their duty if they are willing (my Mom did that and also resigned as trustee that to let me manage things for my dad and for their finances after selling the house, when she was no longer able) and if they are clearly unable the next person in line ("alternate" in my case) takes over, but if they are still alive it may have to be documented in some way. I was able to do that with just letters from my mom's primary care doctor and a geropsych person and blessedly never had to go through guardianship - I needed that because the bank was more strignent about certain things than just relying on the POA I had for for her. Your situation may vay, and again, if it is not obvious from reading the POA documents and/or Instructions to Trustee or anything else you have, you may need to get someone to help out legally, informally as I did with the bank or via consultation with a lawyer. Some of them will at least do an inital free visit - look up "eldercare attroney" and ask around for referrals.