There are a lot of guides in this forum about single-champion strategies, and some of them do mention good laning partners, but there is very little focus on overall champion synergy. I suggest we change this. I am opening up discussion here for what you think some of the best champion combinations are. Preferably 2-champion combos, and be sure to mention both in-lane and late game potential. I'll start:

Morgana & Nunu
These two champions seem made for each other. Both are tankish with a way to self-heal, allowing them a lot of lane staying power given enough mana supply (which can be easily solved with one or both having Clarity + mastery). Their skills also synergize in excellent ways.
Ice Ball + Dark Binding - Ice Ball's Slow makes it easier to land the Skillshot Binding
Ice Ball + Tormented Soil - Ice Ball's Slow keeps them on the Soil longer if Binding isn't available to do so.
Dark Binding + Absolute Zero - Binding can keep champions on the fringe from escaping the Zero.
Absolute Zero + Tormented Soil - Absolute Zero keeps them on top of the Soil for longer
Absolute Zero + Soul Shackles - Absolute Zero provides a stronger Slow to keep the champions in range for the second hit of Shackles.
Black Shield + Absolute Zero - This right here is the cinching point. The biggest problem with Absolute Zero is getting it to last the full 4 seconds against a team full of disables. Black shield provides that ability, giving spell immunity for longer than the duration.

there are a lot of combos. heck, any 2-man group is a combo. the problem is how well they work together.

for instance, eve sucks with most champions. if you want to play her well, you need specific champions that are good at singling out your target and are bad at pushing.

in my elo range, half the time they care and half the time they don't. it always makes a difference who lanes with who though.

items matter too. such as when i get mana manipulator for alistar and my laning partner happens to be kat. i go qq.

This thread comes up every other day, I swear. And every time it's made, the thread is basically 10 posts saying they love Blitzcrank.

Alistar and Blitzcrank. If a Rocket Grab is successful, it is gg. If Blitz is having Grabbing troubles, Alistar can Flash -> Headbutt to Blitz. Blitz can Rocket Grab for some damage and a stun. Blitz pops him up and Alistar comes in with a Pulverize. What are you going to do against that?

Shaco and Janna. Shaco gets Fortitude Elixer and FULL CRIT DAMAGE RUNE PAGE and gets level one Decieve, Ignite, and Exhaust. (Janna could get these too) Janna gets her damage shield. You can hit around 90 Attack Power and 55% Crit Damage AT LEVEL ONE. Shaco gets shielded, Decieves in, gets one backstab. If they are not dead from the one shot. Auto-attack, Ignite, OR Exhaust should be able to easily finish. This works until about level 6...unless you deny the enemy so many levels that you can push the tower with it.