I typically see this as trying to find niches that game giants or other popular games do not cover.
This is the key. As an indie developer, your one and only edge over a large corp is also an extremely powerful one, you have a much greater ability to take creative risk and develop something unique.
When you work for a large company, what motivates the vast majority of people is keeping the steady paycheck. Yes, even in a game company filled with creative people, after a few years some of the big company bureaucracy mindset that is so common creeps in for most organizations. You don't want to be the producer who tries something unique, and it fails; even if you keep your job, you may very well get demoted or at least lose a lot of prestige and be known as "the one with that looney idea that flopped and cost us millions!" So, you stick to what is safe; ideas that are far less likely to be home runs or particularly innovative, but are also much less likely to lose lots of money.
As an indie developer you have more room for risk taking, mostly because your company is still relatively small and flexible. You still are worried about your paycheck, but there is a more entrepreneurial feel that gives you much more flexibility.
Then, as a hobbyist developer, you have the most room for risk taking of all. As a hobbyist developer, the cost for you failing is so low (since by definition games aren't your primary source of income, so it is pure opportunity cost you are risking).
The key is coming up with something that has at least some unique feature or innovation and running with it. That doesn't mean you have to create an entirely new genre or paradigm in the way that Ultima created the RPG genre, id Software created 3d fps, EverQuest/WoW for MMORPG, Minecraft for sandbox etc. That is of course awesome, but far more commonly you can still become very successful within a normal genre with a couple innovative features.
In fact, I would ask myself "is my innovation so unique and daring/risky that major studios would avoid it?" The space where they refuse to operate in is exactly the space where you should be operating. This is a space where on the downside, it is more risky and you should expect a lot more projects to fail, but it is also where home runs and serious breakthroughs are more likely.
Innovation does not have to be limited to game design itself (though that is what myself and probably most people here are interested in). It could be a marketing innovation, in much the same way affiliate marketing propelled the success of so many early internet businesses across many industries. It could be an HR innovation, where you discover a very effective way to successfully recruit talented high school or college game developers. It could be a teamwork innovation, where you create a new communication style that significantly enhances team productivity.
Opportunity to take innovative risks is your one and only edge as a hobbyist/indie developer-- milk it!