How can I prevent my iPhone's messages from showing up on my son's MacBook?

Stop sharing an Apple ID, most likely.

Some background on: How to “unshare” an Apple ID - Apple Community

Make sure that each family member has a unique Apple ID as suggested in this support article --> Make sure that each family member has a unique Apple ID - Apple Support

For a child (age varies according to country), read this support article about how to create an Apple ID for your child --> Create an Apple ID for your child - Apple Support

This support article describes how to create a new, non-child Apple ID = create an Apple Account. --> How to create a new Apple ID - Apple Support

Apple has no formal account merging or transferring process. The things below will help you in doing ongoing or one-time transfer of as much as possible from one account to the other, but some things can never be transferred (namely purchases), only shared.

You can use various iCloud sharing features for some ongoing data sharing.

Share folders with iCloud Drive --> Share files and folders in iCloud Drive on iCloud.com - Apple Support

Read How to use Shared Albums in Photos on your iPhone, iPad, and Mac - Apple Support about how to share photos.

Sharing music playlists —> How to share an Apple Music playlist on your iPhone or iPad - Apple Support

Apps and media bought with one ID are permanently associated with that ID. Look into using Family Sharing to continue to be able to use items purchased with one Apple ID with other Apple IDs, and for some data sharing features.

- Family Sharing - What is Family Sharing? - Apple Support

- Set up Family Sharing - Set up Family Sharing - Apple Support

The next section may sound tricky but it lets you merge some things on a device where you have already signed in with an old account but want to use a new account and incorporate data that is already on the device from the old account. Incidentally, the data will then propagate to other devices using iCloud with that new ID. (I haven't tried this with a computer but it may also work with a computer account signed into an old ID.)

If you have signed into the first ID on a device then fortunately when you sign out of that ID it asks you if you wish to retain the data from the old ID on your device (toggle the ones you want to on). If you keep data on the device then when you sign in with the other ID it will merge this data with the account signing into the device. You can later delete individual items from the old ID that you do not want. Similar information can be found in this post: I have 2 apple IDs i want to merge them h… - Apple Community , and a step-by-step in this User Tip: How to “unshare” an Apple ID - Apple Community . Note that only content physically present on the device like contacts, photos and videos will be retained. Meta data such as a game play scores that are tied to a specific ID or are only on a vendor's Apple ID-specific database remotely will not be retained. Also turn off storage optimization before doing this. With Optimize Storage, smaller, space-saving photos and videos are kept on your device while all of your original, full-resolution versions are stored in iCloud. Make sure photo optimization is turned off so the full quality versions from the old account are actually on the device before merging the content into your new account. https://support.apple.com/HT205703 You can turn optimization back on again once you are fully using your new ID on the device.

Even though apps may remain on a device and may work for a while, you will not be able to update them or redownload them unless you buy them again with this ID or download them with this ID using Family Sharing.

Sign out of iCloud on your iPhone, iPad, iPod touch, Apple TV, or Mac --> https://support.apple.com/HT208242

If another Apple ID is signed in you may need to enter the password for that ID to sign out of that ID first.

Click on this link to learn how to 'Sign in with your Apple ID' --> https://support.apple.com/HT204053