This is refering only to the standard Text message app. No idea how the others are, but with the limitations imposed by Samsung, probabaly the same way. Ok.. the way it USE to be done was through the Contacts Setting. Now you have to start with an existing conversation.
Open the conversation.
Click the 3 dots upper right corner. Click "Custom notifications"
Turn on and click again.
Under Sound, click and pick your notification sound.
Select any other features required such as blink the LED, or vibrate...
Now.. NEVER EVER EVER, Delete that message thread. Because that will delete that custom modification.
NOW also, if you include this person in a GROUP message, and they respond, you will NOT get that custom message. Instead, you will get the STANDARD text message. Since you are not customizing the PERSON, you are only customizing the CONVERSATION.