The word that describes a movement across (over) a border, towards the speaker is:
over, across, ... (towards the speaker)
And in southern parts of Germany and in Austria there is also a very similar word that doesn't describe the movement, but the result of this movement:
over here, on this side
This word is rarely used in northern parts of Germany, but very common in southern parts (Bavaria, Austria). Here are examples for the usage of both words:
- Gestern kam Kurt zu uns herüber. (Kurt came over to us yesterday.)
Kurt moved towards us. When he did, he crossed some kind of border, which can be a river, the border of the township, maybe the doorsill, or maybe just the imaginary boundary around a group of people.
- Heute ist Kurt bei uns herüben. (Kurt is with us today.)
Today Kurt is with us. In the past he stepped over some kind of border (which in most cases is an imaginary border) towards us, and now he is within our areal. (He is not "over there", he is "with us".)
For this word herüben also exists an old shorthand version: hüben. The words hüben and herüben both mean: on this side of the border (whatever the border may be).
So, if you have a word that describes the state of being on this side of a border (in Latin: cis), you also need a word for being on the other side of the border (in Latin trans). It would seem logical if this word was hinüben (because the movement across a border away from the speaker is: hinüber), but language is not always logical. In fact the word was: darüben.
So, the old word darüben meant the opposite of herüben, and is also was transformed into a shorthand version, which is drüben.
- Long versions: herüben, darüben
- Short versions: hüben, drüben
But from the pair herüben - hüben only the long version survived, and it only survived where it was created: in the southern parts of the german speaking region.
On the other hand, from the pair darüben - drüben the short version survived, and it even spread across all regions where you speak German. The long version darüben became completely extinct.
Btw: The word hüben (short of herüben) found a way to survive, but only in one fixed phrase which is hüben und drüben or hüben wie drüben. It means "on both sides".
As far as I found out, the naked preposition or adverb üben without any prefix never existed. Only herüben and drüben exist. (hüben exists as part of a phrase, and darüben maybe existed, but became completely extinct). But from the meaning of the existing words you can conclude that üben (if it existed) would mean "being on one side of a border".