I get this question occasionally and I'm interested in how my answer changes over time. But the core of the answer will always remain the same: to understand how people work. It might be odd that I didn't go into psychology or sociology or philosophy or economics or neurobiology instead, but I found that the magic circle games create around their players and how they chose to express themselves through its mechanics to be more revealing in the ways that I care about. I will admit that part of the motivation of this comes from a lack of common sense and social norms due to quite a sheltered and lonely childhood, so I am eager to make up the difference through my work.
Games that I believe are successful designs are those that are able to create an engaging narrative arc (doesn't need to be a good story, but could be the progression of a player's own abilities), have a wide possibility space derive from a simple player actions, and have thematic cohesion in every aspect of its construction (from sound design to the choice of menu font). These are probably things that you have heard before from much smarter people or perhaps you might disagree with them, in which case make your case in the comments. There are three folks that I look to when thinking about design: Yasumi Matsuno, Mark Rosewater and Supergiant Games.
Yasumi Matsuno (YazMat) is a Japanese game designer and writer on titles such as Ogre Battle, Final Fantasy Tactics, Vagrant Story and Crimson Shroud. I came to know his work through one of the first games I owned and still go back to time and time again: Final Fantasy Tactics Advance. It was my first introduction to the world of Ivalice which remains one of the most detailed and wondrous that I have encountered in games. In my opinion, he is a master of low fantasy plots with large casts of characters and political intrigue. The English localization (done often by translator Alexander O. Smith) of his games have often taken a Shakespearean tone that elevates the work to an almost mythic level. The game designs are heavily systems oriented making them have a large learning curve at the beginning, but around mid-game players will understand enough to experiment and make their own ways of playing the game. My big takeaway from his work was to not compromise on one's vision (many feel Final Fantasy XII was full of missed opportunities) and build one's audience by commitment to aesthetic. YazMat's games a treasured rarity in any game library.
Mark Rosewater is an American game designer who has worked on the card game Magic: The Gathering for over 20 years. Rosewater is best known for making the game accessible to new audiences by adapting psychographics (Timmy, Johnny, and Spike) and narrative with story arcs and characters (such as the Planeswalkers) into its creative design. He is also an avid blogger and podcaster which has helped the community understand a lot of the decisions that go into Magic's design. I also highly recommend checking out his GDC talk: 20 Years 20 Lessons. Rosewater emphasizes his studies as a Communications major as being a critical part of his design process. One of his design pillars is the concept of lenticular design that any given element of your game should have an easily understandable use for new players and then as they gain experience that element is re-contextualized to have higher skill function. He is not a figure without controversy as many pro-players in the Magic community have railed against his initiatives (New World Order being a prominent example), but he is also the first to admit when a mistake has been made and tries to give good reasons for the decisions he makes.
Finally, we have Supergiant Games who are an indie game development studio formed by Amir Rao and Gavin Simon in 2009. Playing their first game Bastion was a massive turning point for me because that is when I realized that I wanted to pursue game design as a career. Bastion's brevity of language and purposefulness in every aspect of their design made it stand out from its competitors. There are many indie games can flaunt beautiful visuals, great music and robust mechanics but few can pull them together into a cohesive thematic whole that is a memorable experience in the way Supergiant does. Their second game Transistor shows their amazing toolbox design of interchangeable parts while their most recent game Pyre is a triumph of worldbuilding, character writing and incorporating loops of victory and defeat seamlessly into the narrative. They represent the high bar of game making that I constantly aspire to and I hope to work for them someday, even in a limited capacity.
Hope that gives you all a better idea of what I am looking for in my designs. Whenever I'm in doubt, I ask myself what these folks would do. Sometimes I get different answers and that's fine because game design is a tricky thing where we must discover the answers by getting feedback from players and breaking through our own assumptions.