A lot of the hip-hop artists that use references to the mafia culture are typically involved with crime (or at least try to sell themselves as such). The mafia culture--the most well-known form of organized crime--has always been a guilty admiration of Americans, white and black alike. In the '70s, '80s and '90s, it grew more and more popular and appealing, thanks to various movies (The Godfather Trilogy, Scarface, Casino, Goodfellas, Carlito's Way, etc.) created throughout the years. And what music genre was developing during the same period of the release of these movies? Hip-hop. A lot of the aspiring hip-hop rappers of yesterday saw a lot of parallels between their own lives and what they saw in the movies, although to different degrees: drug dealing, robbery, homicides, rivalries, money laundering, etc. Obviously most rappers who were actually living the "street life" weren't as organized as the mafia, it was something that they aspired to be, much like how young adults trying to be entrepreneurs in the computer world look up to Bill Gates. So perhaps that is why that it is mostly rappers and not rock-n-roll or pop singers who, in their music, assume the character of a don, capo, underboss, button, and so forth.
However, the use of mafia references is usually either an exaggeration or metaphor in reality. Tupac, who seemingly was the most successful at using the mafia image (especially as a marketing tool), obviously wasn't a mob boss, but he was the leader of a rap revolution and had connections with Suge Knight, who himself had considerable pull with several Blood gangs and is arguable the person involved in the rap game who is most like an actual mob figure. Unfortunately, after Pac's death, it became the cool thing to do... not just to be a thug, but to be a don, a mob figure. So following Pac's death, we saw a lot of clones who couldn't pull off the mafia image as successfully or as believable... you can find tons of examples of such in Master P's No Limit Camp and all of their various aliases and album titles... Master P's MP Da Last Don, 504 Boyz's Goodfellas, C-Murder's Bossalinie, Silkk the Shocker's Made Man, Gambino Family's Ghetto Organized, Young Bleed's My Balls and My Word, etc. So unfortunately the whole mafia thing has been completely saturated, even though references are still made here and there.