I have a strange question. Could this sentence be correct?

Hi evanescence, and welcome to the site.

Ask your friend what "he" means in this sentence:

He who laughs last, laughs best.

In view of the fact that "one" was used in the beginning of the sentence, I suppose the use of "one" and "one's" would create a better balance in the second half of the sentence. 

On the other hand, using "one" in the beginning and then "he" and "his" in the second half of the sentence might be looked at as something the author did for stylistic effect -- i.e. start off with a very generally stated question (in essence, about "anyone") and then finish with words that more strongly suggest a particular male person.

Nowadays, politically correct language discourages the use of he/him/his to refer to a person of unspecified gender since the person might end up being female.  However, this sort of use of masculine pronouns is a traditional usage.

Both "one" and "he" can be used as gender-neutral pronouns, as can the word "you", for example.  The use of "one" often seems more formal.

Having said all that, even if "one" had been used throughout the sentence, I suspect it would have still been abundantly clear to all involved exactly who "one" referred to.

That's my two cents.


EDIT:
PS
To me, your second sentence refers to a particular male -- i.e. the usage is gender-specific, not gender-neutral:
How can he be taken seriously if he can't even spell his screen name correctly?

"One" and "he" can be used as genderless pronouns, and they can be mixed together in the same sentence too. I personally hate "one" because it sounds way too formal to me... at least in American English. Is it just my impression? How likely is someone to use "one" in speech? Anyway, I usually just don't use "one" and I say or write "a person" or "someone" instead.
As for genderless "he", I personally avoid it as well, but it's not a matter of being politically correct. It's just that it's odd, because "he" and "him" are automatically connected to "a guy" or "a man" by my brain, and perhaps that happens to native speakers too. Just consider this example:

Go out on the street, approach the first person you see, and kiss him. <--- No thanks. Not for me. Amy might have no problems with this, though.
Go out on the street, approach the first person you see, and kiss them. <--- This is better. This version allows my brain to think I can skip the guys and just approach the ladies.

LOL So it's often just odd. However, it's still used that way, and still commonly found in formal writing. 
That's my opinion on genderless pronouns.