What's this syntax called: "Him being the nice person he is, helped her."?

You offer two options for handling the sentence that you are interested in:

Him being the nice person that he is, helped her.

and

He, being the nice person that he is, helped her.

But there is a third option:

His being the nice person that he is helped her.

and it may help you make sense of the difference between the two options you originally suggested.

Your use of one comma in the "Him..." sentence but two commas in the "He..." sentence indicates that you see at some level that the phrase "being the nice person he is" behaves differently in the two original sentences. Crucially, in the "He..." sentence, "being the nice person he is" need not be anchored to the spot where you have placed it. It works every bit as well at the front of the sentence:

Being the nice person that he is, he helped her.

or at the end of the sentence:

he helped her, being the nice person that he is.

Quite clearly the same is not true with regard to the "Him..." sentence:

Being the nice person that he is, him helped her.

and

Him helped her, being the nice person that he is.

are constructions that no moderately fluent English speaker would ever use. Why? Because in the "He..." sentence, the phrase "being the nice person that he is" is an independent clause and therefore has the ability to float to different positions in the sentence—immediately before he, immediately after he, or immediately after her—and still make sense. That's also why it makes sense to set commas before and after the phrase, as you did in your original sentence:

"He, being the nice person that he is, helped her."

But in the "Him..." sentence, "being the nice person that he is" is not independent. It is physically tied to the pronoun Him: The subject of the sentence isn't merely "Him" but "Him being the nice person that he is." This is where the "His..." alternative may offer some insight. As written, the wording "His being the nice person that he is" is equivalent to "The fact that he is the nice person that he is." If this fact helped her, we can express that meaning by writing the sentence with no commas:

His being the nice person that he is helped her.

This is a legitimate English sentence, but it clearly means something different from "He, being the nice person that he is, helped her." In the latter sentence, he helped her because of who he is; but in the former sentence, the fact that he is who he is helped her.

In colloquial English people often use him for his, meaning that you could use

Him being the nice person that he is helped her.

as the equivalent to the "His..." sentence, with the same meaning. And if you wanted to delve a bit deeper into colloquial speech, you might say

Him being the nice person that he is, he helped her.

with a meaning equivalent to the more generally approved form

Being the nice person that he is, he helped her.

English speakers say things like that all the time and everyone understands them, even though, strictly speaking, the Him at the beginning of the sentence is superfluous and arguably unjustifiable except through common usage.

But to return to your original question "He, being the nice person that he is, helped her" is perfectly correct English with the meaning "He helped her because he is a nice person"; "Him being the nice person he is, he helped her" is marginally okay as a colloquial way of expressing the same idea; "His being the nice person that he is helped her" is perfectly correct English with the meaning "The fact that he is a nice person helped her"; and "Him being the nice person that he is helped her" is a generally accepted somewhat colloquial way of expressing that same idea.