From Jack Weatherford's The Secret History of the Mongol Queens:
"The daughters of Genghis Khan bore the title beki, an honorific designation applied to either a prince or princess. The men who married them, however, did not receive the title khan or beki. They received the special title of guregen, generally meaning “son-in-law,” but in this case the meaning was more akin to “prince consort.” A man held the title only because of his marriage to the Great Khan’s daughter. If he lost her, he lost the title. Because the guregen could be so easily replaced, the chronicles often merely used the title rather than the name. For most practical purposes in daily Mongol life, it mattered little which man was actually filling the post at the moment. If one died, another quickly stepped into his place. Usually the replacement would be a son, brother, or nephew of the last husband.The guregen occupied a unique position within the Mongol imperial system. Despite the high prestige of his close kinship to Genghis Khan, he rarely held any influential military or civilian office. Genghis Khan kept the guregen literally close at hand; most of them received appointments within the keshig, the royal guard, and thus became intimate parts of Genghis Khan’s personal camp. Those with superior ability became the leaders of their own military units, composed of about one thousand warriors from their own tribe or related clans, but under close supervision and never too far away from Genghis Khan. In this way, the best warriors in the guregen’s tribe were always separated from the majority of their tribe. A guregen had no chance to rise up within the hierarchy and no chance to wield independent power or launch his own campaigns. He held a prestigious but hardly enviable position in Mongol society.The guregen served under their father-in-law, according to the tradition of bride service that Genghis Khan had revitalized and strengthened. Instead of herding the father-in-law’s goats, camels, and yaks, these sons-in-law became herders of men; they would serve in Genghis Khan’s army and fight in his wars. Genghis Khan often sent his guregen on the most dangerous missions, however, and they tended to be killed at a high rate. Most of them never had the chance to return home for very long. Being a son-in-law to Genghis Khan was not merely an apprenticeship phase, as it would be for a small herder; for these men it became a brief, but usually lethal, career.A tribe acquired prestige and material benefits if Genghis Khan chose to marry one of his daughters to its leader. For the son-in-law, however, the honor was almost certainly a death sentence as well. He served virtually as the sacrificial victim in exchange for his tribe’s prosperity. He would give his life in battle for Genghis Khan, and in return his tribe would benefit and his own offspring would be rewarded. A guregen entered into a harsh bargain."
Perhaps GJ made the right choice by opting to become HYS' son-in-law instead.