A little Moon
What is the little Quartermoon beside the name in my contacts list and how do I get rid of it?
iPhone 11, iOS 14
Question marked as
Swipe left in Messages for that contact and tap the Alerts button. See the following from Use Messages on your iPhone, iPad, or iPod touch - Apple Support
Turn on Hide alerts
- Open Messages.
- Swipe left over the conversation that you want to mute.
- Tap the Alerts button
.
When Hide Alerts is on, a crescent moon icon 
will appear next to the conversation. This stops notifications only for that conversation, not your device. You’ll still receive all other messages and see notifications for them on your Lock screen.