The struggling hobby that is e-Sports has gone through a lot of ups and downs recently, so this is certainly an interesting question to pose: what does make a good e-sports game?
The first thing that will come to many gamers minds will be the type of game it is. First-person shooter fans will be quick to defend the FPS genre, along with real-time strategy fans for RTS (and of course, the sub-genres within these genres for one-versus-one or teamplay, different rule-sets and such). However, I believe this division is what is holding e-sports back from becoming truly mainstream.
There have been some leagues - albeit none taken very seriously - that have tried to cater to this division in utilizing a multi-game format approach; however these have mostly been for fun or very small cash prizes and thus not gotten a lot of coverage or interest.
The most successful e-sport currently is StarCraft, and that is largely because the playerbase has refused to deviate from this title; thus allowing spectators ample time to learn and understand the game - and for the players to become extremely skilled at a single title rather than spread their skill thin as other e-sports players have to do quite often. It will be interesting to see what StarCraft 2 does to this e-sport scene, as real sports don't have to deal with sequels, however I think the impact will be minimal as it has grown so large as to be a part of the culture itself in the main StarCraft hub, Korea. It is in every way, shape and form an "electronic sport".
Overall, I think the biggest thing that a good e-sport title needs is consistency, which is mostly up to the players to determine. Everyone has their favorite games, but a decision needs to be made on a lone title to use as the platform for that specific e-sport genre. As we saw with the CPL, CGS, the old cyber games and other leagues rise and fall, the reason mostly seems to be due to the inconsistency with games being played and the lack of staying power. As with any sport, it is up to e-sports to have a good fan base to generate the revenue necessary to run them. With most leagues constantly bouncing from game to game, fan interest is very hard to hold and they seem to inevitably fail. The game needs to be simple to understand, yet have a large amount of skill and depth involved. It needs to have a good level of showmanship for the all-important spectators, and there needs to be a pre-existing level of fandom for the game itself - you cannot simply make a game specifically for e-sports and expect players to embrace it.
Given all of the aforementioned attributes, and the fact that game developers rarely seem interested in e-sports, I'll have to chalk the answer to this question up to luck. As Korea became infatuated with StarCraft (due to it being one of the first titles released internationally there), it become more and more a part of the culture to the point of where it wouldn't be considered strange to hear two people talking about it on the street. Although it is not strange to hear people talking about videogames in other countries, they are often talking about the topic as a whole, and not a specific game. Indeed, it is rare to even find someone with the same preference in games as you in most of America and Europe. If a game becomes truly explosive in popularity in these regions, and boasts the necessary traits to become an e-sports title, we may see resurgence in this hobby in these regions. Until that day comes, whether or not you can make a living off of professional gaming, I think e-sports will still largely be considered a hobby.