The honest and less helpful answer is that it depends. It depends on the champion and depends on the designer. I'll talk about only the reworks I was involved in and that actually went anywhere, which were Jax, Trundle, and Sion.
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The primary motivator for the Jax changes was the removal of RNG dodge as a mechanic. Jax being the primary user of dodge had basically lost his E ability and a large chunk of his thematics if we simply stripped this mechanic out. The rework was geared around how do we keep this champion intact and still feeling like himself.
Shurelia had the great idea of "Is dodge the problem? Or is it just RNG dodge the problem?" since we agreed it was the latter 100% dodge chance seemed like a great solution and the rework commenced :)
Trundle was mostly an art driven rework. The creative and art teams really wanted to update him and this was a good opportunity for us on design to pop in and do some minor tweaks to go with the art rework. We were happy with Trundle's kit for the most part already so the changes were geared towards increasing satisfaction (the rationale for the impact slow on Q, bump on E, and stronger visual support for the ultimate/Q) and more clearly identifying his role as a tank buster (changing the Dmg on R to be % based like the armor/mr steal), this is why the Trundle rework as a whole was a pretty small changelist overall despite the large creative shift.
Sion has enough problems designwise, artwise, and creatively that in most ways the rework is basically a new champion, and we're using a very similar approach and style to when we create completely new champions. We'll have more information on him as he moves along but we're approaching it from a full rework perspective and we don't feel there is a ton of the current character worth keeping so he will be very different once complete, unlike the previous 2 champions I mentioned who we worked very hard to retain as much as possible. In Sion's case if we can retain elements of the current character from art/design/creative it's certainly a plus, we just don't see much worth trying to keep on the current design. It's not quite starting from scratch - we still want big hulking zombie champion from Noxus, but it's certainly the same level of work as making an entirely new champion.
