There are a couple of things you can do.
You can simply give your son your log-in information, which gives him access to all of the functionality you have. You can add multiple ship-to addresses to your account as well as multiple credit cards. Basically he could log into your account, use his credit card and his shipping address, and enjoy all of your benefits. The reason many people do not do this is because then the other person also has access to *your* credit cards, meaning he could purchase things and charge them to your credit card and you would have no recourse with the credit card company because you gave him this access. You might trust your son completely, but be aware that it's easy for a spouse or child to get in this way too -- especially with the password saving conveniences of modern computers.
The other way to add him is to create a household account. This means that you each have a different Amazon account, but you link the two accounts into a household that will share in your Prime benefits. Amazon used to allow many adults to share, but in 2015 they changed household accounts to allow 2 adults and 4 children. Here are links that explain what a household account is and how to set it up:
https://www.amazon.com/gp/help/custo...deId=201806360
https://www.amazon.com/gp/help/custo...deId=201806380