If you want to make a favourites list that’s available for all clients you can make a m3u playlist containing your chosen stream URLs and save it in musicbox’s /var/lib/mopidy/playlists directory. Finding the URI for what’s playing has become a little trickier lately but you can see Adding default Radio Streams for one way of doing it.
As an example, I have the following saved in /var/lib/mopidy/playlists/Favourites.m3u
#EXTM3U
#EXTINF:123, BBC Radio 2 (London)
tunein:station:s24940
#EXTINF:0,Absolute Radio (London)
tunein:station:s16679
There might be a hacky way to make it appear first in the list of playlists shown by the webclient but I don’t know.