The simple interpretation of the phrase "Yeheh Shemeh Rabba Mebarach" – which we respond to Kaddish – is "May the Great Name be blessed." Meaning, we declare that "Shemeh Rabba" – G-d's great Name – shall be blessed and glorified. Tosafot (Berachot 3a) cite this interpretation in the name of Rabbenu Yishak.
However, Tosafot also bring a different interpretation of this phrase, in the name of the Mahzor Vitri, according to which this phrase actually consists of two separate proclamations. First, we declare that "Yeheh Shemeh Rabba" – that G-d's Name shall be great, or complete, and we then add that it should be blessed forever ("Mebarach Le'alam…").
The concept underlying this second explanation is that G-d's Name is "incomplete" in our nation's state of exile. G-d's complete Name consists of four letters – Yod, Heh, Vav and Heh – but when the Jewish People are in exile, His Name is compromised, consisting of only Yod and Heh. This is indicated by G-d's pronouncement after Beneh Yisrael's first battle against Amalek: "Ki Yad Al Kes Y-ah" (Shemot 17:16). As long as Amalek exists, Hashem's Name is only "Y-ah," missing the final two letters, Vav and Heh. For the same reason, G-d refers in this verse to His throne with the word "Kes," a shortened version of the word "Kiseh," indicating the incompletion of His reign, so-to-speak, due to the presence of evil in the world. The Maharsha (Rav Shmuel Eidels, 1555-1631) explains on this basis the verse in Tehillim (132:13) which we recite each morning, "KI Bahar Hashem Be'Sion, Iva Le'moshab Lo" (literally, "For G-d has chosen Zion, desiring it as His abode"). This means that when Hashem will choose to return to Zion, with the rebuilding of the Bet Ha'mikdash, then "Iva" – the letters Alef, Vav and Heh – will be restored to their rightful place ("Le'moshab"). The missing Alef of the word "Kiseh," and the Vav and Heh missing from Hashem's Name, will finally return. Similarly, we recite each morning after Shema, "Hu Kayam U'shemo Kayam Ve'chis'o Nachon" – "He exists, His Name exists, and His throne is firmly set in place." Rav Shlomo Amar explains this to mean that when "Hu Kayam" – the letters Heh and Vav will come back, then "Shemo Kayam" – Hashem's Name will again be complete, and "Ve'chis'o Nachon" – His throne will be complete, as well. This notion is alluded to also in the prayer added by Ashkenazim to Birkat Ha'mazon: "Ha'Rahaman Hu Yishlah Lanu Et Eliyahu Ha'nabi" – "The Compassionate One – He will send us Eliyahu the prophet." When Eliyahu comes, "Hu" – the letters Heh and Vav – will be restored.
Another allusion to this concept appears in the final verse in Tehillim: "Kol Ha'neshama Tehalel Ya-ah" – "Every soul shall praise Y-ah." The word "He'neshama" ("the soul") can be punctuated differently such that it means "the desolation." This verse thus implies that in the current state of "desolation," in the absence of the Bet Ha'mikdash, we can praise only "Y-ah," as Hashem's Name remains incomplete.
According to this second interpretation of "Yeheh Shemeh Rabba Mebarach," we are praying that "Shemeh" – a contraction of the words "Shem" and "Y-ah," referring to the incomplete Name – shall be made "Rabba," complete, and, additionally, His Name should be blessed ("Mebarach").
Incidentally, the word "Yitgadal" consists of five letters, and the word "Ve'yitkadash" consists of six letters. These two words thus represent the letters Heh and Vav, which, respectively, equal in Gematria 5 and 6. We pray through the recitation of these words that G-d's Name should be glorified through the restoration of the missing Vav and Heh.
According to this second opinion, when reciting "Yeheh Shemeh Rabba Mebarach," one should pause slightly between the words "Shemeh" and "Rabba," as this is a prayer that the Name ("Shemeh") should become great ("Rabba"), and the word "Rabba" is thus not describing the word "Shemeh."
The Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909) cites the Arizal as concurring with this view of the Mahzor Vitri. However, as Hacham Ovadia Yosef cites, Tosafot challenged this reading in light of the Gemara's comment (there in Berachot) that speaks of Jews proclaiming in the synagogue, "Yeheh Shemeh Ha'gadol Meborach" ("May His great Name be blessed") – which clearly indicates that we are declaring that the great Name shall be blessed, as Rabbenu Yishak explained. Moreover, Rav Yisrael Bitan notes that the Ben Ish Hai elsewhere brings Rabbenu Yishak's understanding, and ruled that one should therefore not pause between the words "Shemeh" and "Rabba." This is, indeed, the correct practice to follow.
Summary: The accepted interpretation of "Yeheh Shemeh Rabba Mebarach" is "May the Great Name be blessed." One should recite these words without any pause in between the words "Shemeh" and "Rabba."