Is there a way to display a player name in /tellraw command?

Tellraw allows for a "selector" part, which takes any selector you want and displays a name. You should do:

/tellraw @a {"text":"","color":"blue","extra":[{"selector":"@p"},{"text":" has won the game!"}]}

This will only work in 1.8 snapshots and above. There is no way to do this in older versions.