The "o" at the end of this word is a suffix that means "his." I purposely use the masculine form of this suffix (rather than the English gender-neutral "its") because what matters in this word is what the antecedent of "o" actually is.
"Shem" means "name." And where Judaism differs with Shakespeare is in the essential nature of the name. A rose by any other name may smell as sweet, but we know that a rose is a rose because it is a rose. Gertrude Stein was no more correct when she said, "a rose is a rose is a rose." Names matter greatly in our tradition. God gave the first human being, Adam, the ability to name the animals because Adam was able to discern the essential nature of each one, not because he had a flair for creativity.
Shem follows "l'ma'an" in this blessing, which means "because of" or "for the sake of." So whose essential nature makes it necessary for God to "bring a redeemer to [our ancestors'] children's children?"
The answer must be (grammatically speaking) a preceding noun that is masculine and singular. Therefore, it cannot be "our ancestors" (plural) or their "pieties" (also plural).
Possibility number one is God's own self. Since God's attributes are the major focus of this paragraph, you might reasonably assume that God brings a redeemer for God's own sake. While it makes a certain sense from a literary point of view, it does not sound quite as logical with a larger perspective.
Possibility number two is the redeemer himself. To be called "go'eil" is to have a name that conveys a certain essential nature. Therefore, you might argue, the redeemer will be brought because otherwise he would not be called the redeemer. However, such circular reasoning seems a little foolish.
The third possibility has to do with an idiom in Hebrew, usually expressed in the contraction of the two words "l'ma'an sh'mo" (or the feminine form). To speak of "torah lish'ma" is to talk of learning Torah for its own sake. L'ma'an sh'mo, or lish'mo, means simply "for its own sake." God brings a redeemer just for its own sake.
Why? That question is answered by the last word in the body of this blessing.