Mechanics being the most important is a general rule of thumb that most games follow. However there are great examples of games like Gone Home and Life Is Strange that buck that trend (In those cases Story, Music and Mechanics would be the proper order). Music clearly follows Mechanics in terms of importance in my opinion. Music can make or break any moment in a video game regardless of it's genre. The storyline could have been excellent, the graphics could be perfect, but if the music is off it ruins the whole thing. This is why Tekken Tag Tournament for instance stands from other fighting games to me because of it's superb soundtrack. Story follows Music in importance but with a huge caveat. Many genres like fighting games, platforms, puzzle games don't need stories at all (In those cases Mechanics, Music, Level Design would be the proper order). Story is very important for narrative driving games however.