Many of us know elderly men and women who no longer act as we have come to expect them to act. I'm not talking here about victims of senile dementia. In the example I am thinking of the person continues to behave in what most people would agree is a normal manner, but one so remote from his old self that he appears, to those who know him, to be someone else entirely.

About the 'one' above, which does it refer to; 'a manner' or '(a) person'?

I think it's 'a manner', but I'm not sure on this one.
I agree - a manner so remote...
The pronoun one (not the one) refers back to an indefinite, some noun with a or an.
The antecedent would have to be a manner, not the person or most people.

Henry needs a new pen -- one that doesn't leak ink in his pocket.
*Henry likes most people -- one that is very friendly.