So I've been reading a lot of lore articles about TES, and it seems that each Daedric Prince has a specific date they can be summoned on (some have other times they can be summoned as well). This would be pretty fun to have in the next game, I think. Instead of finding a hidden shrine, or speaking with a random NPC, you should have to wait until a specific date (or event) to summon them on your own, wherever you are. Doing Daedric quests this way, you could also call them forth once a year (or more, again depending on the Daedra) just to shoot the breeze

I don't think summoning days would work with the current setup for TES games. They made sense in Daggerfall, because in Daggerfall you literally burned in-game days and weeks fast-travelling, waiting a year wasn't so bad as long as you paid attention to the in-game clock and budgeted your time. (You could also literally spend in-game days in a single dungeon, with mages settling down for naps when their magicka ran out, for example, or warrior types sleeping in the dungeon when their health was low, or simply being lost in the dungeon for days) In the modern days of Morrowind-Oblivion-Skyrim, you simply don't burn that much time doing anything. The Main quest of Oblivion can beaten before you ever get to Frostfall, as long as you don't get too distracted with side quests or guilds, and the game starts pretty close to the end of Last seed. In Daggerfall, meanwhile, your character will spend months of time merely fast-traveling during the MQ.
Unless they change up the game mechanics to allow more frequent passage of large amounts of time, summonings days would have to be more common in order to work as a game mechanic.