The bit where Abraham asks for God's name sounds like they are still strangers to each other having spent little time discussing things, and yet God selects him. Why is that?
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I suspect you may be confusing several different verses.
For starters, thinkers such as Maharal have argued that the Bible omits most of Abraham's backstory because it's not our business to know about G-d's criteria for choosing him. "Why are the Jews the chosen people? Because G-d chose them." The focus is instead on doing our part, now that we are chosen.
Now in Exodus Chapter 6, G-d gives Moses (several hundred years after Abraham) a history lesson and tells him that He was not known by the name YHVH to Abraham, Isaac, and Jacob. This means that they never had the full prophetic experience that Moses would achieve, and that they didn't get to see the "beyond-time" existence of G-d that Moses would. (Abraham, Isaac, and Jacob all got promises about what would happen to their children; Moses got to see those promises carried out.)
Or perhaps you mean Exodus 33, where Moses has pleaded that G-d not destroy the people for worshipping the golden calf. Successful, Moses goes on to try to better understand G-d (which our commentaries say includes understanding G-d's ways; both forgiveness for bad things, and bad things happening to good people). But the only way to know the mind of G-d is to be G-d, and we are human and finite, so Moses goes farther than any other human with his success, but still only so far.
You don't need to know God's name to know that there is a god whom you should be paying attention to. There is a midrash in Bereshit Rabbah 39 where Avraham deduces the existence of God who created the world, and only then did God call out to Avraham. Is there a causal relationship? The midrash implies it but torah never tells us; as @Shalom said, the torah itself doesn't explain why God chose Avraham (or Moshe or Yisrael), just that He did.
Summary of the midrash from my notes after a class:
So, to sum up, God chose Avraham because Avraham was perceptive/curious enough to seek out the Master of the Universe.