The word that really concerns us here is "bein" ("ben"),* the plural of which is "banim." By removing the prefixes and suffixes that mean "to" and "their," we are left with the phrase "b'nei banim." Idiomatically, it means grandchildren. But literally, it means children's children, that is, succeeding generations.
The word itself has three meanings. The one with which we are most familiar is "child." As you know, Hebrew allows the masculine to also be the generic, so it would be accurate to say that "bein" also means "son." It accounts for our familiarity, for when a man or boy is called by his Hebrew name, it includes his given name, followed by "ben," meaning "son of."
The word can also mean "member." The fraternal service organization B'nai B'rith has nothing to do with patrilineage. "B'rith" means "covenant" (in the lisp pronunciation of Hebrew; Ashkenazim say "bris" and Sephardim say "brit"). A Ben B'rith is a member of the covenant community, not the child of the covenant. Similarly, we use the Aramaic equivalent of "ben" when we refer to a bar mitzvah. The young man has a mother and a father, but no mitzvah has shared parenting responsibilities. Instead, he has become a member of the community of mitzvah-observers.
Finally, "bein" can mean "worthy." When we refer to someone as a ben-Torah, his learning is noteworthy. Gentiles who are called b'nei No'ach are those who adhere to the seven commandments given to the generation after the flood.
All of these meanings are present in the phrase at hand – offspring, members, deserving ones. The use of the plural form that directs us to the future is the arrow that points us from this realm of history (that began with our focus on the Patriarchs and Matriarchs) to the future. It illustrates what William Faulkner said (many centuries later): The past isn't past; it isn't even over yet.
We are the b'nei banim of our forebears, as are our children and their grandchildren after them. When God promises to bring a redeemer "to their children's children," an unbroken chain is affirmed from the very beginning of our history as a people to its eventual fulfillment.
*Bein/ben is spelled bet-nun, not to be confused with bet-yod-nun, pronounced similarly but meaning "between."