It seems like it could be a lot of fun. There are certainly quite a few design and technical challenges to overcome though.
A lot of champions have abilities that are individually pretty bad alone but have additional effects when combined with another of their abilities. For example all of Brand's abilities have a bonus effect when combined with his passive. His Q especially is pretty meh if you don't have the ability to get the stun effect. Similarly, it might be possible to get Cass's Twin Fang and no poison for resets, or to roll Yasuo's ult but no displacements on the team and not ever be able to cast it.
How much of an issue this is depends on whether you're rolling abilities from the entire champion pool, or just those of the 10 champions that were selected in each game. Even so, Karma and Syndra's passives become non-functional without the abilities they're tied to on the same champion, and Karma and Heimer's ults also don't do anything without the right corresponding abilities.
Another potential pitfall is the transform champions. Their ultimates don't do much besides flip from ranged to melee without the champion having the other abilities they're supposed to have. And if you rolled Jayce Q for an ability, do you get To the Skies! or Shock Blast? One potential solution is to assign additional Q/W/E abilities to transform to if a transform ultimate is selected. So if you roll Jayce's ult, you could get Fiddlestick's Terrify in one form and Sona's Hymn of Valor in another.
The biggest issue may be how to deal with different resource systems. Most champs use mana, but there are several champs that spend energy instead, and some that spend no resource (or only health) to cast their spells, and a few other scattered resource types. Do Lee Sin or Katarina get to use mana abilities? What if you roll both energy and mana abilities on the same champ? This one I don't really know what to do with other than either eliminate non-health costs and resources entirely, or try to convert all abilities to mana costs and give all champions mana pools.
Still I would love to see the hilarity if you start with Caitlyn and roll Trists's Passive, Jinx's Q, Kog'Maw's W, Kayle's E, and Twitch's Ult (that's up to 1888 range at level 18.)
