Please note that the term "Events" may appear in place of "Activity", depending on the version of Remote Services you're using.

Entries in the Event log can be flagged with a as ‘favorites’. Events flagged in this manner can be searched quickly using the star icon in the filtering options on your mobile device or PC.
Starred favorites cannot be saved beyond 90 days.
 
To mark an Event as a favorite:
 
Touch the  button next to an event to mark it and it changes color:  . 
 
Note: Video events cannot be starred as Favorites. However, you may save them permanently to your device with the Download icon .
 
This example is taken from the iOS app.