You can change the notification sound in general GroupMe settings, and if you're on Android or Windows you can also set a different notification for each individual chat.
To change the general notification sound:
iOS: Make sure you've enabled notifications for GroupMe in the iOS settings app. Tap the menu icon and select Settings, then under Notifications, tap Sound and choose a sound from the list.
Android: Tap the menu icon (top left) and select Settings, then under Notifications, tap Sound and choose a sound from the list.
Windows: Tap the menu icon (top left) and select Settings, then choose a sound from the list for Group Message Sound and Direct Message Sound.
To change the notification sound for a specific chat:
In the chat, tap the avatar and select Settings (or Chat settings) then choose a sound from the list. This is only supported on Android and Windows.