I think that giving Alizée a kiss on the cheek is more a present for you than her.
Not that getting a kiss from one of her biggest fans is not a great gift, it's just that most of the French always exchange cheek kisses between each other when greeting friends, or when they are introduced to a friend's friend. So if you can ever arrange to have yourself introduced to Alizée by one of her friends you'll probably be able to get and give your cheek kisses.
And back to the topic, I have to think a bit more before I can decide what I would give her...