The Maharitz Chiyos in his Toras Hanevi'im (in the note there) wonders how Reuven could seriously make such an offer. Is one's child's life a commodity? He suggests that perhaps Reuven's ability to make such an offer was pursuant to his status as leader of his brothers and presumptive king of the nascent Klal Yisrael.
The Chasam Sofer in his last teshuva in Orach Chaim, written to the Maharitz Chiyos, disagrees, and says that this was just Reuven's way of saying that he would give up his din bechor, his claim to a double inheritance. "שני בני" means his bechor status, which gave him a claim as if he were two heirs. He explains that the expression "תמית" is just another way of saying that he would give up his claim to land in Eretz Yisrael. The Gemara in Bava Basra 118b says that owning land in Eretz Yisrael is referred to as חיות, life:
Rav Baruch Epstein says this pshat also, and he makes it clear that his way of putting it is superior to the Chasam Sofer's.
In Parshas Vayechi, Yaakov gives a bracha to the sons of Yosef. In 48:4, it says ונתתי את הארץ הזאת לזרעך אחריך אחזת עולם. Then, in 48:5 it says שני בניך הנולדים לך בארץ מצרים עד באי אליך מצרימה לי הם אפרים ומנשה כראובן ושמעון יהיו לי. Finally, in 48:16 it says ויקרא בהם שמי. We immediately realize that these two sons of Yosef have thus supplanted the sons of Reuven- they were counted as separate shevatim, thus granting a double portion to Yosef. Now, the Rov brings from the Rashbam and the Ramban that ויקרא בהם שמי means שיחיה זרעם וזרע זרעם, or שיעמוד זרעם ושמם. He explains that passuk 4 shows that the essence of the Bracha was inheritance of the land; 5 shows that this bracha gave Yosef's sons the status of being Yaakov's sons and thereby independent Shevatim, who are therefore entitled to individual portions, thus yielding for Yosef a double portion. The Gemara in Bava Basra 115b states אמר אביי גמירי דלא כלה שבטא, we have a kabbala that no Shevet will be entirely wiped out. Also, he brings that the Toras Kohanim Bechukosai 8 says ברית כרותה לשבטים, and the Raavad there says ברא הקב"ה ברית שלא יכלה זרעם. According to this, we can see how meduyak the Chasam Sofer's idea is in the words of Reuven and Yaakov: If Reuven's sons would have the status of Shevatim, this would yield two results: double portion of land, and a guarantee that they and their names would survive forever. By waiving his claim to the Bechorah, Reuven exposed his children to the possibility of death. We don't need the Chasam Sofer's connection of the word חיו to ownership of land. The חיות at issue was real חיות, the guarantee of survival, and that is why Reuven said את שני בני תמית.
The Aruch LaNer, in his Binyan Tziyon, says he doesn't like the Chasam Sofer's pshat, (although I don't think his complaints are very strong,) and suggests a different pshat. He says that Reuven was making an oath that he would bring Binyamin back. The Gemara (e.g., Shabbos 32b,) says that transgression of an oath can cause the death of minor children. At that time, two of Reuven's four children were minors, so he was telling Yaakov that he was willing to make a Shvu'ah that he would bring Binyamin back, even knowing that failure to fulfill the oath would be fatal to his two minor sons. With a little imagination, the Aruch LaNer's pshat can be read into the Targum Yonasan, who says ית תרין בניי תקטול בשמתא אין לא איתיניה לוותך.
What I find most interesting here is the apparent difference between the perspectives of Reuven and Yaakov about the relationship between a person and his grandchildren. Reuven held that the connection of Yaakov to Reuven's children is contingent on Reuven. If Reuven's relationship with Yaakov would be destroyed, then his children would no longer have anything to do with Yaakov. Yaakov held that Bnei Banim K'Banim, that the relationship to a grandchild stands independent of the relationship to the child. Obviously it stems from that first connection, but it acquires independent significance. Even if a child were disinherited and estranged, the relationship to the grandchildren would not necessarily be affected. It could be that this is the Shakla v'Tarya of the Gemara in Yevamos 62b, where it's pashut that בני בנים הרי הן כבנים as far as the son of a son, but it's a chidush that it applies to the son of a daughter. It also comes up in the din of yerusha b'kever in Yeish Nochlin in Bava Basra 116b, as follows:
So it turns out that this question is at the bottom of Rami bar Chama's iboya.
The question also arises in matters of family law- in many, if not all states, grandparents have an independent right of visitation with their grandchildren, except where the exercise of such rights infringes on the superior rights of the parents. This should be tolui on the machlokes between Yaakov and Reuven, the Gemara in Yevamos, and the iboya of Rami bar Chama in Bava Basra.