"Well" as an introduction to an argument
Say a child says:
I want some ice cream!
The parent's response is:
Well, you can't have ice cream right now, we need to have dinner first.
Why is the word "well" used as a conversational introduction to an argument? Is it a shortened form of another phrase?
Solution 1:
Well might originally be a shortened form of a phrase, but I think it is safe to say that people who use it are generally treating it as a distinct entity (not short for something); it is a grammatical particle that is used to "facilitate discourse".
In this situation, it is being used by one speaker to acknowledge what the other speaker said. It can also be used to additionally indicate "I'm about to say something now, so wait a moment and don't say more until I've taken my turn."
We have many discourse markers, and in general they are used to assist turn-taking and comprehension in a conversation. Every language (as far as I know) has them.
Solution 2:
It may be a vestigial form of "that's well", "that's all well", "that's all well and good" and similar sentiments, which in the exchange you describe would indicate that it's perfectly all right that the child wants the ice cream, though it isn't going to be happening right now. Other uses of "well" as an interjection don't fit this pattern, but the one you're asking about, a conversational introduction to an argument, does.
This is, in fact, how Wiktionary understands "well" as an interjection used to acknowledge a statement or situation to have derived: as short for "that is well".
Solution 3:
The OED offers no etymology for this use, but only the following definition:
Employed without construction to introduce a remark or statement, sometimes implying that the speaker or writer accepts a situation, etc., already expressed or indicated, or desires to qualify this in some way, but frequently used merely as a preliminary or resumptive word.