Advertising for big-budget action games like Call of Duty and Halo, some outdated conventional wisdom, and some misguided editorials might have led you to think that video games aren't for you. If so, don't get stuck there. Consider an analogy: maybe today's top-selling pop song isn't your favorite (on November 19, 2014, that's a tie between Taylor Swift's "Blank Space" and Hozier's "Take Me to Church"). That doesn't mean that the entire medium of music is one that you can't enjoy. To find music that you like, you might have to range a little further than full-page advertisements in newspapers and prime-time commercials on television. You might instead discover the music that appeals to you through asking friends for recommendations, reading papers with smaller distributions (and blogs), patronizing small clubs, and browsing independently owned music stores.
Likewise, the games that you'll enjoy might not be the ones with the largest marketing budgets. Ask your friends what they're playing--you might be surprised by both who answers the question and what they answer it with. Investigate the indie game scene and search some online game stores (such as Steam and itch.io), which, unlike retail mall stores, aren't exclusively stocked by publisher-funded releases.
I list below some of the great games made in the last few years. These have a breadth of gameplay, themes, and visual styles. They're all intelligently crafted. Some, such as Monument Valley and Minecraft, are actually the best-selling games on their platforms despite a lack of coverage in mainstream media. Others, such as Papers, Please, have small sales and are niche finds. Not all games are about narrative, but the games on this list that tell stories present ones that are not in the generic Hollywood style. They have diverse protagonists and those protagonists do interesting things. Many of these games are suitable subjects for a thesis.
If you're seeking for games for children, apply the same judgement that you would for film and books. Research a game ahead of time and play at least part of it with the child.
Just because these games are great doesn't mean that there's anything wrong with also playing the games in the Halo series, just as there's nothing wrong with listening to Taylor Swift's music. Large, mass-market productions are of course enjoyed by many people, and a benefit of that economic scale is tremendous polish. I enjoy a lot of pop music and a lot of AAA games. I'm also proud to have contributed a very small amount to the large team that built Call of Duty to the premier gaming franchise. Such AAA games are among the most complex human endeavors, representing the latest engineering and person-centuries of effort towards a perfected play experience for their target audiences. Yet, I'm also proud of the indie games to which I've contributed--those present individual, uncompromising experiences. They push the boundaries of games as both an entertainment and art form, and welcome different audiences.
I chose the games mentioned in this article for diversity, accessibility as an entrance to the medium, and their appeal to me. There are hundreds of other great games like them, and new games are released daily. I hope that at least one of them opens games to you. Through these I became a war hero, a starship captain, a trans woman, an abused child in a favela, and a British adventurer. Those human experiences were otherwise inaccessible to me.
Just as I also enjoy many mass-market games, you may find that you like those too, and have been mislead by what you've heard about them. It would be silly to let someone else's opinion preclude your own evaluation. The only mistake in this space is assuming that any particular work in a medium defines all of the content within that medium, and that mistake seems to be made frequently with the particular medium of video games.
Please start looking at some of the games on my list, talking to friends about less-advertised games, and seeking reviews of "indie", "casual", and other games that aren't covered by national publications. The great promise of a game is that it is limited only by the imaginations of the developer and the player. Many of today's mass-marketed games tell the story of a particular demographic, but there's room for everyone's experience in games and many of those experiences are available today. So, start enjoying a medium that is open to everybody, including you.