Sounds like you've already been giving him the best gift of all - your time and yourself! Maybe make it a day that he gets to pick what he'd like to do, with you and/or a good friend, is my immediate thought.. the next few years will be full of great change, big and small, so tangibly, a father passing on his shaving kit or a new one, or a mother letting her son shop for his own clothes, for example, can also be a gift of themselves as they're signs that they know how hard teen years can be so they're letting him start to take new steps towards being a man by learning to make his own choices (not saying you don't.. just an example!), learning new skills, and being more independent.
Having 5 of my own, from 19 to 1, your question has been something I've struggled with.. but I've learned that it's all based on your own individual child and what, to them, would mark this time in their life as a new chapter. My oldest son wasn't always interested in a thing i could give him but how I looked at him in our conversations, discipline, responsibilities, allowances, etc. from then on. And it doesn't have to be right at 13.. again, it's all unique to each child. Some still seem like they're 8 when they've turned 13 and aren't ready for anything to change. But again, the gift of your love, time and understanding (especially if they suddenly would rather share things with their friends than you), is the best you could ever give him.
Though I really love the book idea too! Adding some blank pages at the end or presenting the first book of memories with a second one, left blank to be filled with future memories and accomplishments of his teen years, will show that you miss and cherish the little boy he was and the memories you've made, but will also be cheering him on with pride and confidence as he grows to discover who he has yet to become! Though, by the sounds of how close you two are, he probably already knows that. :)