My opinion:
- It does not matter if the partner is fast or slow. They don't need to match in every aspect, and what really matters here is that the partner can take certain hits. Togekiss is an excellent spreader of paralysis. Perhaps the Pokémon itself could get Thunder Wave, in order to free Togekiss's moveslot? However, that brings me onto my next point.
- We should be wary of creating something that is so good as a supporter that people will be more inclined to use it independently of Togekiss.
- The Pokémon should be neutral or resistant to Rock. I agree with those who are saying Stealth Rock could defeat the entire purpose of this duo. If we want them to succeed then they should not require too much outside support. This also helps Togekiss. Alternatively, the partner could carry Rapid Spin, but I think creating another good Rapid Spinner will make it more of a standalone-used Pokémon. It also doesn't help as much vs Tyranitar.
- The Pokémon should be resistant or immune to Electric, or otherwise be able to solidly beat the OU Electrics. Jolteon, Zapdos, and Rotom-A are the main threats to Togekiss. While Ice-type attacks generally don't threaten Togekiss since it has Roost and the only threatening STAB Ice-type attack in OU comes from Abomasnow, the STAB Electrics in OU are far more problematic and also resist Togekiss's mainly used STAB. I'm sure we can all agree Electric-type attacks are one of the main problems for Togekiss.
- The Pokémon should have a weakness to something that Togekiss can cover. This provides some mutual synergy and encourages people to use the two together.
- The OP asks 'How far should we go when with synergy? Will too much focus make our Pokemon too reliant on Togekiss's presence? Will too less not even affect her effectiveness?' The idea here is to make a 'perfect mate'. In my opinion, this means we should err on the side of too close synergy. At the very least, it will be a boost to Togekiss. However, we should obviously try to strike a happy medium. I think when we've reached a point where 'Togekiss and this Pokémon function best alongside each other but do not need to be alongside each other to work in OU' (as obvious as it sounds), we've succeeded. I think we need to ensure, in this point anyway, that our CAP is okay on its own. There's nothing we can do to Togekiss itself within the realms of this CAP. If we want it to work best in tandem with Togekiss rather than other Flying-types, perhaps we should 'capitalise' upon Togekiss's Normal typing by giving it problems with Gengar? (Note: LO Focus Blast 2HKOs Togekiss with Stealth Rock) However, this would also give it Rotom problems. Perhaps it should have its own anti-Ghost attack and fall in between them in speed tiers. That's just my thought, anyway.