There's this particular interjection-like usage of there that I think belongs to the spoken register and maybe doesn't refer to anything; as in:
- Hi there!
- You alright there?
Why is it used, and what meaning (if any) does it serve to convey?
Hi there! is a greeting. There refers to the position that the other person is in, so it is an adverb. It can also serve to attract attention.
Hi over there! or Hello over there works the same way as Hi there or Hello there, except that the distance between the two people is greater. The two people could be across a room or across the globe.
I think you you are right about social register, as Hi there! seems to me to be informal or at least not formal. This is also indicated by Hi.
The there in You all right there is also an adverb. You can also say things such as Hey there! and You there! and not say it in a friendly way. You can also say Who goes there?
Another stock phrase is
There she goes!
where, again, there is an adverb and is used to attract attention to whatever she is, be she a female person, a boat, or, in this case just about anything.
With respect to people here who still think that "there" in greetings doesn't refers to distance here is my explanation:
Firstly, educated people (mostly native speakers) say that before, when sailors wanted to draw the attention of someone at a distance (to hail a ship or anything else at a distance) they would shout, "Ahoy, there!" in which "there" refers to a distant object. Sometimes "there" was followed by the identity of the object: "Ahoy, there, ship/captain/matey".
Now we use "hi", "hello", "greetings", "hey" and etc. instead of "ahoy" but "there" is still used to emphasize the physical distance between the speaker and the object addressed.
Often "there" is used in place of the unidentified object when the object can't be visually identified at a large distance of because of fog, mist, or other visual obstacles.
"Hi/Hello/Greetings/Hey, There" is not impolite or anyway rude, but mostly appears in informal usage.
Footnote: "there" isn't the dummy subject in these greetings. "There" is a dummy subject mostly in constructions there is or there are, or before certain verbs in certain contexts.
'There' in such sentences is used to indicate direction or place where the person is and it is informal. It directs attention towards the person indicated.
Some interesting responses here about "hi, there," although I would argue that in common usage, no matter its origins, it's mainly a cheery colloquialism.
On the other hand, if it helps, I'd most likely use a question like "Are you all right, there?" to address someone who seemed spaced out (distant in thought, in other words).