In the cut-scene for Half Life 2: Episode 2 Gman says he "plucked" her from Black Mesa. Why?
Speculatively, GMan is 'in control' of the timeline of the Half Life series - certainly in the opening speech he gives in Half Life 2 he gives the impression it's him pulling all the strings. There are more references in his final speech of that game.
Given this, it's probably for the same reason he 'extracted' Gordon Freeman (or led him to safety, depends how you read his words from that speech I referenced earlier). I'd say it's because Alyx is somehow needed to save the world.
Perhaps it could be said that GMan is somehow a 'protector' of the world, and that Alyx and Gordon are chosen as heroes for specific abilities they possess. Alyx certainly seems to have an affinity with technology (see 'Dog' character from the series, and her proficiency with the gravity gun).
I think this is really as deep as one could go without resorting to making stuff up - it's already highly speculative and based on pretty thin strands of evidence.