The Blood of Vongola is what every Vongola Boss must possess in order to be the true and rightful owner of the Sky Vongola Ring. This bloodline starts from the Vongola Famiglia's founder and first boss, Giotto. If one doesn't carry his bloodline, the Ring will reject them. A good example of a rejected boss candidate is Xanxus, who tried to become the 10th Generation boss, but failed when the Ring rejected him.
The Blood of Vongola also grants a blood trait called Vongola Hyper Intuition that gifts them perceptive mind and sharp instinct.