How about we use the WoW Battleground model for featured gamemodes?
For those who don't know in World of Warcraft there are several pvp Battlegrounds that you can q up for. When doing so you can either choose a specific one you'd want, or pick random battleground and get bonus honor (pvp currency) and every weekend a specific bg will be featured and also grant bonus honor.
So I thought that system would work pretty well for featured game modes, since riots main argument against making them permanent is that it would split up the q's too much and take forever to find a game. That way if you'd just want to have some non serious games and you're sick of Aram you could pick random and get some bonus IP outta it or pick a specific one you really want to play (and let's be honest, it's U.R.F.) accepting the longer wait and no bonus ip.
Ip earnings from featured gamemodes are usually pretty low anyway, so it wouldn't screw with the in game economy.
The biggest problem I see with this is that some players might get butthurt if they don't get the gamemode they wanted and leave/afk, but they would be sorted out by the leavbuster pretty quick and not cause any problems for too long, I would think.