For someone who plays a video game for the story there is a very definite line between a good story and a GREAT story. This depends a lot on the character development. As we play games we are escaping into another reality and replacing our life with the character in the game. When the character development is done exceedingly well this transformation allows the player to connect with the game on a much deeper level. Maybe the how the character handles death, forms a bond with other AI, or demeanor changes over the course of the game can have a major effect on the player and overall game.
Last of Us
The relationship of Joel and Ellie is a major player in the Last of Us. Joel goes from feeling as if this girl is a burden and he is taking her as part of a job to make money. As the game develops and the characters are put into worse situations the dialogue and tone of the conversations there is a change in Joel's attitude toward Ellie. This very subtle continuous change over the course of the game has a major impact on how the game is received and the player engagement.
In the recent Tomb Raider reboot, the player controls a Lara Croft that is very fragile and has not killed anyone yet. The player connects a lot with the character of Lara Croft because we see every single time that she hurts and kills, developing the character.