About that whole 2 or 3 for each role thing, I'm pretty sure you just need supports and you'll be fine. If you don't want to mid or top, you're not going to have a hard time getting someone else to do it. Jungle and ad carry maybe, but solo lanes are REALLY popular.
Well, actually, I have the whole champ choice thing sorted out, especially in the bot lane. Besides having a handful of ADCs I can use well, (Ezreal, Caitlyn, Vayne, Varus, Ashe) I've got at least five or six supports I can use efficiently enough, especially since I started to use melee supports well.
Bot lane duo are a pair, you have to match your partner, so you need a tank (vayne/tristna), a poke (ez) and Nunu (caitlyn). Worth it if only to deny your opposing carry their best support f they don't pick together.
What I have as far as these support types go would be Thresh and Taric as tanks, Lux, Lulu and Sona as pokes, and Nunu as... Nunu. Since I took Ryuu's advice on melee champs, I have been playing Taric and Thresh in the support lane like crazy, with mostly positive results. I'm still looking to increase how well I sync with my lane partner, though.
For solo lane, you want a roster such that there does not exist a champ who counters all of the ones you have. So if you have 4 top champs but they are all countered by teemo, that is a bad idea. If you have 4 mids but they are all countered by Talon, that's a bad idea, too. Luckily if you pick sufficiently different champs, this situation should rarely occur. For instance, for mid it should suffice if you have a long-ranged mage with poke and utility, a mobile assassin and a tankier/bruisery guy, methinks. For top you'd want a tanky/bruiser guy or two plus a harassing bastard.
Additional variety can never hurt though. Also, there's several champs that can be played effectively in multiple positions (Cho can mid, top and jungle, and he is cheap, for instance, kayle can literaly do every role, though she is better in some than others).
Yeah, I try to keep one hell of a lot of variety and keep to learning champs that work well in several different lanes. Kennen, Hecarim, Malzahar, Teemo... I can play most of them in multiple lanes effectively. But overall, my mid and top lineup goes as such:
Mid - Kennen, Lux, Katarina, Malzahar, Talon, Elise
Top - Kennen, Malzahar, Hecarim, Teemo, Vi, Elise
I have more, but these ones cover the variety mentioned pretty well. Not sure about a tanky bruiser down mid, though.
Really, the only reason I listed that first point in the "why I don't go into ranked" is because when I stopped playing ranked, I felt one of my problems was lacking viable junglers and supports. Fixed that, so now it's just the other two things I'm looking into.
Hecarim & Elder Lizard nerfs
;-;