It all depends on what you mean by "save".
If you just want to save the message as pure text, copy and paste it into a text file and delete the message thread.
If you want to "keep that one message in your texting app", you cannot delete the thread. As Rukbat suggests, you need to delete the individual messages that you do not want to keep.
Depending on the texting app, you might be able to "forward" the message to yourself and see if that starts a new thread for it, or email it to yourself.