The LED of the message key on registered handsets shows when a new message has been received. This function is called Message Waiting Indication (MWI).
Via the Web configurator in Settings - Messaging - Message Notification you can for each handset individually set on behalf of which type of new messages the LED should flash. The message types are:
For each handset select the message types because of which the message key shall flash when a new message has been received. To do this, click the option next to the message type. You can assign several or no message types to each handset.