I believe players would enjoy this little detail in the interface. When hovering over the different player icons, a little text with the name of the icon could appear. This way the player would remember where the icon originates from. For example: "Doom Bots 2016"
To be honest, not the most practical use of time and energy. Plus, I feel that I've read somewhere that Summoner Icons were meant to create personal experiences for players (like a way for players to express themselves). Icons can be personal, I just don't think they are for most people.
But, having names for icons or titles is something everyone can appreciate, in a really small way, like percentage of games that are Urgot games small.
I think it would be extremely time consuming for Riot to go back and name every single icon. I'd rather they focus and work on things that are actually important/related to game and client fixes than naming icons.
