How do I save/lock specific comment bubbles within a text message thread? Is it possible to make the downloaded SMS application, "Google Messenger" save just part of a thread in a text conversation on my Moto E Android phone? 

Example: I'm chatting back and forth with a friend, a long thread, a long conversation.  In one comment bubble, my friend gives me an important address that I want to lock and save, but I want to delete the rest of the thread (all the other bubbles).  

I used to be able to do this on an older android phone.  It had a different sms app.  I don't remember what that app was, but I could delete my entire inbox with the exception of specific comments/bubbles on which I had put locks/saves. 

My current phone model number is Moto E (2nd Generation) with 4G LTE.

OS is lollipop 5.1 (I think)
I think you've answered your own question in your post when you said that you used to be able to do this, but had a different SMS app.

This function, as far as I know, is app specific. I am able to lock messages using ChompSMS on Android 5.0.

Try different SMS apps and see which is best for your needs.