Kerstin here from the Shopify Guru Team :)
You can do this with the Gift Wizard app which is probably be our most popular app for adding a gift button to your product page. And an alternative would be Easy e-Gift Checkout. You might also be interesten in the Shopkeeper Gift Cards Shopify Gift Card App which allows you to send customized gift cards.
All the best,
Kerstin | Shopify Guru
Any updates on this from Shopify? It seems like this should be an option within Shopify vs a $10 charge a month simply for gifting. Thoughts on if Shopify will include soon.
Our new Application, GiftShip, might be the perfect fit. It comes with many powerful gifting features, including the ability to send items to different locations within the same order, and attach personalized messages per item as well.
Check out a few stores that have recently brought on our functionality:
Let me know if you want to chat further about it :) We offer free installations and trials, and have different pricing plans to suit your needs.