In the Kaddish, we say that may G-d be praised "beyond any blessing, hymn, praise, and consolation said in this world."
I understand how we can bless or praise G-d -- but how exactly (or even fuzzily?) do we console G-d?
Based on Dave's comment, so hat tip to him:
The Rokeach here says that the nechama (consolation) we provide is on G-d's regret for destroying the Temple, as He missed out on the nice singing that would occur if the Temple remained extant. As such, he recommends that the prayer leader lengthen the word V'nechemata", and think of consoling G-d and consoling us (presumably those praying or the Jewish people in general).
I have heard in the name of Rav Hirsch (though I lack a specific citation) that nechama can be a change of mind, not only consolation. This Rokeach seems to fit that meaning, in that G-d changes His mind from destroying the Temple to, if it is possible to say, wishing that He had not done so, similar to: וַיִּנָּ֣חֶם יְהוָ֔ה כִּֽי־עָשָׂ֥ה אֶת־הָֽאָדָ֖ם (and G-d regretted having made man).
Excerpt from this beurei hatefilah article:
Summarizing the concept, Talmud Brachot (3a) states that G-d is saddened when he realizes that He has exiled his children from his house. The following is excerpted from the end of the article (English translation section):
When the congregation then comforts G-d because of G-d’s sadness the language switches back to Aramaic so that the angels will not understand that the congregation is comforting G-d. Everyone knows that the purpose of comfort is to relieve sadness. Then the congregations says: OO’L’Aila Mikal Birchata Yitbarach and OO’L’Aila Mikal Shirata Tooshbichata Yishtabach. The words: OO’Mikal Nechemata are meant to comfort the sadness that G-d feels in His world.
No source, just my thoughts, but here goes. The Kaddish has a strong association with mourning. One way we comfort mourners, especially through Kaddish, is by reminding them that HaShem has a plan and that their suffering isn't for naught. This can be a great source of comfort to a mourner, and that itself is a great praise of HaShem. As such, the praise of HaShem's consolation is a fitting threshold for comparison for just how much HaShem should be praised (especially in the Kaddish itself).
Kaddish is about the redemption of Israel and the restoration of God's kingdom. The Hebrew/Aramaic term "Nechama" (translated as consolation) means to reconsider, to mentally fix something. The redemption of Israel and the restoration of God's kingdom is the greatest possible "nechama" that can be had.
The Artscroll book Kaddish has several answers to this question. One is that the word וְנֶחֱמָתָא may be a synonym for mincha, a gift/offering as per the targum to Yoel 2:14. So the list of praises includes God's name being greater than any praise that could be offered by way of gift.
Another answer is that the list of praises are all referring to various types of prophecy, one of which is nechama. The point of the list is that God is greater than anything any of the prophets were able to describe (even in prophecies of nechama).