I realize the question is worded strangely. I'm listening to Dan Carlin's Hardcore History Wrath of the Khan's series. He talks about succession of power and I get that; but it seems to be limited to only three or four sons.
My question stems from the fact that he apparently had so many children that you have a better chance of being related to him than almost any other figure in history. I realize some (probably most) of his children were left with whatever people he had just raided, but I'm sure he had more than a couple who traveled with him. Did he consider them all his children? How did he pick the three or four to succeed him?
His "successors" were his children by his first wife Borte: Jochi (predeceased him, may not have been Genghis' son, what would have been his khanate was split between his own sons), Chagatai, Ogedei, and Tolui, with Ogedei as the khagan.
I would direct you to this thread: How realistic is the Genghis Khan genetics theory?
Also, /u/76vibrochamp is correct. Genghis Khan had four successors: Jochi, Chagatai, Ogodei, and Tolui.
Jochi - Died in 1225, before Genghis Khan. Jochi, roughly translated, means guest. His mother, Borte was kidnapped by the Merkit tribe in retaliation for Genghis Khan's father, Yesugei, kidnapping his mother, Ho'elun, from the Merkits. His sons were Batu Khan of the Blue Horde and Orda Khan of the White Horde. These together form the Golden Horde. If you've never heard of the Golden Horde, they were the scourge of the Kievan Rus! The Golden Horde dominated Russia until 1481. The last Crimean Khan was dethroned in 1783.
Chaghatai- Formed the Chaghatai Khanate. It extended from the Amu Darya to the Altai Mountains and remained active until the 17th century. Half of the empire was taken by Tamerlane in the mid-1360s. The Mughal empire of India claimed they were descended from Chaghatai.
Ogodei- He was the Khan of Khans after Genghis Khan died. His son Guyuk became the Khan of Khans after his death.
Tolui - His son Mongke became the Khan of Khans after Guyuk Khan's death. His second son Hulagu became the leader of the Ilkhanate in Persia. The Ilkahn territory was dissolved in 1335. Kublai Khan became the Khan and established the Yuan dynasty in China. His descendents ruled until 1368. Kublai was never the Khan of Khans. He killed his brother Ariq Borke because two great meetings, kuraltai, were held and Kublai and his brother were both elected Khan. Kublai could not unite the steppe after this incident.
Sources are from Jack Weatherford's Genghis Khan and the Making of the Modern World, Francis Woodmen Cleaves' translation of The Secret History of the Mongols, Morris Rossabi's Khubilai Khan: His Life and Times, and the chart below used in a graduate course on the Mongols.
Here is a hand chart that shows this information -> Genghis Khan's Lineage
Edit: Formatting and link fix.