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".