As long as we're talking about evolution, it should be understood that this happens slowly through natural selection. At the time when Chess was evolving into Modern Chess, communities were more isolated from each other, and there was no organization codifying the rules for an international community. So, local variations sprung up here and there, and as communities with different local rules came into contact, some rules got favored over others, and eventually there was enough international agreement on which rules to use, and FIDE was created to establish and promote the same rules throughout the world.

In the present, FIDE Chess is dominant, and it has a huge international infrastructure behind it. For another variant to displace it in this present environment would be very difficult. If something like a nuclear war or the Carrington event knocked back civilization, then that might create the opportunity for a new variant to overtake FIDE Chess in popularity. Short of that, the best chance any variant has of overtaking Chess is to build up popularity through tournaments and literature.

Now, in recent times, new things have sometimes been known to overshadow old things in popularity. This is mainly due to emerging new technologies. For example, CDs made records much less popular, and then streaming made CDs less popular. With games, video games have become very popular, and newer video games continue to become more popular than older ones, because they feature better graphics and use better technology. If video games erode the popularity of Chess, and the Chess world becomes more like the video game world with players interested in a variety of games instead of just one, then other variants might have a chance of displacing Chess. One step in this direction is to move the typical place for playing Chess from face-to-face across an actual board to over the internet, and a second step is to make other variants available over the internet on the same sites where Chess is played. What would be really helpful would be to have websites where people can easily tinker with Chess and play their own variants against a computer. This would help lead people from thinking about Chess as a single game with fixed rules to thinking of it as a certain type of game with a variety of possibiltiies to try out.

Given such an environment, I suspect that the games that would do best against Chess are ones that do not veer away from Chess too much. Those that introduce a couple new pieces that fit with the other pieces, such as Omega Chess, Grand Chess, or Eurasian Chess would probably have a better chance than others. There's also a chance that Chess could go in the direction of Blernsball (the future version of Baseball in Futurama), where something like Knightmare Chess, which combines the appeal of card games like Magic: The Gathering and Yu-Gi-Oh with Chess, would do well. In fact, if card games like these overtake Chess in popularity, something like Knightmare Chess could serve as a gateway to the world of Chess and its variants.