The evidence would seem to suggest the D. is a family, not a trait that appears on random births signifying great future. The evidence for this is that we do not know how any characters got their "D." name from, except for Ace, Dragon and Luffy who obviously had it through being sons of Ds. What's more, Dragon calls it "inherited Will", making a strong argument for a bloodline.
Saul might be a half-giant, he is small compared to Brogy or even the Giant VAs in Marineford, which would make his D. as part of a family possible. Even Teach has some traits of characters (like love of food and adventure) that most Ds have.
Now unto my wild theory. Keep in mind i am not CONVINCED this is true, i just speculate using bits of evidence. To read with an open mind.
My wild theory, that COULD happen based on the evidence but is highly unlikely, is that the name of D. is the bloodline of the rulers of the country that the World Government destroyed during the Void Century, and are being used as pawns in a master plan to bring down the World Government as revenge. My reason for believing that is that every single "D." has affected the World Government in a deep way.
-Gol D. Roger was the Pirate King that started the Great Pirate Era, leading thousands of pirates to sail to the Sea, attacking the authority of the WG.
-Portgas D. Ace was the cause of the Marineford War, that ravaged Impel Downs and Marineford, two of the WG's greatest holds, and Portgas D. Rouge was the person that made that happen by being his mother.
-Monkey D. Luffy destroyed Impel Downs, Enies Lobby, brought down 2 Shichibukai, etc.
-Monkey D. Dragon is the WG's most feared enemy.
-Jaguar D. Saul's sudden betrayal made Robin escape from Ohara, making it possible for someone to read the Poneglyphs this kingdom left behind. Her later joining the SHs and being taken by CP9 led to the destruction of Enies Lobby.
-Blackbeard D. Teach released the Level 6 Prisoners from Impel Downs, weakened the Shichibukai Order with his betrayal, half destroyed Marineford and Seabondagy Archipelago.
-As for the Hero Monkey D. Garp, he is the only unfitting link in the chain, but i'll justify it by speculating his role would have been to be the father figure (literally or not) for Dragon, Luffy, and Ace, three major "Ds" whose roles are explained above.
So the Ds caused the destruction of the World Government's three main bases, the Great Pirate Era, and the likely future discovery of the True History once Robin reaches Raftel. Without the Ds, none of this would have happened.
AGAIN, it is SPECULATION. I do not hold it as truth. I just think it is an interesting thing to think about.