That person....their

Sounds too awkward, I'd never say it. If someone else said it to me, I'd do a double take. It just doesn't make sense to use one's in the place of my. It sounds as if you are saying your car is everyone's prized possession. Perhaps they use one's this way more frequently where you are from (whether it be another part of america or another country).

<What if you'd said "My car is one's prized possession."?>

Then you'd be speaking improper English, in my opinion.

Sounds awkward to me too but no less so "I like a person who sticks to one's principles."

Nobody said "I like a person who sticks to one's principles" didn't sound awkward. I am just saying that it is technically correct.

No, it's not even "technically correct." You can't use 'one' like that unless it's referring back to a previous use of 'one' as a nominative.

"One must stick to one's principles" is fine.

"A person must stick to one's principles" is not right. The meaning, if a sentence this mangled can be said to have one, is "a person must stick to someone's -- not necessarily his own -- principles."

There's no good way to use the impersonal 'one' to rewrite the sentence "I like a person who sticks to his principles."

"I like a person who sticks to his principles" is the simplest, cleanest way to express it.

If you must rewrite to avoid (perceived) sexism, then:

"I like a person who sticks to his or her principles" is OK.
"I like a person who sticks to their principles" is not old-school textbook English but it's becoming more and more common colloquially, and will probably be considered acceptable at some point. I don't say it and certainly wouldn't put it in formal writing.

Hmm, I like your explanation. I agree I'd use "I like a person who sticks to his principles" or "their" in place of "his" as well.

I'd have to agree. It'd be technically incorrect whatever you'd wnt to mean by "I like a person who sticks to one's principles."

I say "their" in place of "his" or "her" all the time, and not just when the gender is unknown. I'm not the only one either.