[Acheron]

I have to disagree with complimenting a girl at all, let alone on her eyes. By starting a conversation that way, it creates a frame in which you are saying nice things to her in order to win her approval. That's not a good thing: successful frames are those in which she tries to win your approval.

While it may be appropriate to compliment her later in a conversation, you should still avoid the eyes. Hugely cliched. In fact, avoid anything physical, and focus on something she's done well or some aspect of her personality. Makes her feel like you're assessing her on more than just her looks.

Oh, and don't buy things for girls as an opener. Makes it look like you're trying to buy their affection, and while they might keep you around for being a pushover, they won't respect you.