No, none of the little legends have any impact on a player's performance in a game of TFT. However, if you are a bit superstitious like I am, I do notice when I win streak with a particular little legend, and I tend to consider those my lucky ones. My luckiest little legend so far has actually been the Snow Cherry Silverwing! I attribute some of my wins to that pretty little bird; despite the fact that he really had nothing to do with my gameplay.