Let's list some disparate facts
1 Aveilim say kaddish to raise the level of the n'shama of the dearly departed
2 Aveilim also lead Benching for the same purpose
3 The Talmud highly commends saying the "Y'hei sh'meih" "b'chol kocho"
4 Tosafot [and others] see shmeih as shin meim Yud Heih meaning "the name - Kah" instead of simply shmeh [I.e. sans a Yud] which means only HIS name.
I posted a while back about the minhag of Aveilim saying Qaddish and leading benching [points 1 and 2 above iirc on Nishma Minhag] as being connected because they BOTH elicit "Qiddush Hashem" by the responders
And note while With benching it is the Passuq Yehi Sheim Hashem m'vorach.
Notice that there is an apparently tiny gap
Y'ehi shmei translates technically as y'hi shemo
Whils the passuq in Hallel and Benching is Y'hi sheim HASHEM.
Here's the possible nexus. Perhaps Tosafos has a Tradition that y'hei shmeih IS the Targum for Tehillim 113:2 and that this Oral Tradition is the impetus impelling Tosafos to find Hashem's name [albeit the short version] within this responsem Then the reading that appears as shmeih [Aramaic]meaning sh'moth [Hebrew] really means Shem Hashem.
That would tie the response in Benching even closer to that in Qaddish, and make the common denominator of Qiddush Hashem even more compelling.