For some reason when my son sends my mother a text message it shows up with my name. Texts from me show up in a different thread, but when I click on the "contact" from his it takes it to my contact. I do not use Apple any more (my son does), but he does use the Apple ID I set up when I had my old iPod. The only thing I can think is that is what's causing the problem, but I can't figure out how to resolve it. Any suggestions?
On your mother's phone, delete your Apple ID (it's linked to the email – that's the info you need to remove) from your contact info (the page you are brought to when you click the contact). Add this same piece of info to your son's contact card.
Your son should have his own Apple ID, but if you're not using it anymore I don't think it's necessary for him to create his own. He can change the email address to his own at appleid.apple.com
Alternatively, he could just make a new Apple ID, and you can keep your old one in case you ever get another Apple device. This is honestly probably the easiest solution, unless there is some reason you want him to use the other account.
This was occurring on my iPhone 7 except that it was with non-family members or people outside my same Apple ID or phone plan. It started after a restore from a backup.
I was able to fix it by going into Contacts and clicking "Import SIM Contacts". Then import to iCloud. This lined my contacts, and conversations back up to the correct sender.