It really depends on the player first, the champion second, and the role third. To raise your ELO in soloqueue, you generally want to pick a champion that can make up for a bad or uncoordinated team. I'd suggest looking at the CLG solo queue tier list as a place to start. It isn't word of law by any means (and it isn't current), but it'll give you an idea of which champions can really carry a game.
You want to pick a champion that both lanes well and transitions into late game. I find that champions with an initiation ability also do well because soloqueue teams underestimate the need for crowd control or initiation. Pretty much every champion in tiers 1 and 2 of that list is either a strong laner or strong jungler, and they either bring high dps, good initiation and crowd control, or both to teamfight scenarios.
I would say that focusing only on individual play isn't going to do you any favors. If your team seems decent but lacks direction, try to encourage them to take objectives like dragon, baron, or towers when you seen an opportunity.