Hi there, youremyyooniverse!
Thanks for reaching out to the Apple Support Communities! That is a great question you have about naming your group chat and adding another user and I am happy to help! Based on the screenshot, it looks like you have a contact in the group that is not using iMessages which means they are using SMS and MMS messaging, so I recommend you take a look at this article to Send a group message on your iPhone, iPad, or iPod touch.
For the question about naming the chat:
You can name only group iMessages, not MMS group messages.
For the question about adding another member to the group:
You can’t add or remove people from group MMS messages or group SMS messages.
To add another person to the group chat, you'll need to create a new group chat and include the new member to the group.
Cheers!