"Something to fill up the hole on the heart": Friendship
When Minato said Bee's more of a "true ninja" that has something more special, than a Jinchuriki's power, he's probably referring to his potential to sacrifice himself for those he cares for. I guess Jinchu's are just used to being weapons that use their powers for destruction that it was rare to see one risk it's own life for the sake of another, the way Bee risked his own life to save his brother. Naruto has that same determination and the Rai-man should also see that in him and realize that if Bee was in Naruto's current situation he would also be risking his life to save his brother as well.
Up till now the Raikage see's Naruto's actions as nothing more than the irrational notions of an immature child but this flashback was more for us to learn the reason the Raikage will probably FINALLY understand Naruto's determination. Not just for wanting to risk his life in order to join the battlefield with his friend but also when he confronted him on the way to the summit to plead for Sasuke's life. Naruto and Bee are the same so if he kills Naruto then he WILL have to kill Bee as well, cause Bee wouldn't let that shit happen.