Based on R' Samson Raphael Hirsch's commentary on Shemot 6:2-5 and his citation there of Isaiah 52:6, I'd like to suggest that the ultimate Redemption can be identified with attribution of God as the Author of everything, and that perhaps, this concept motivates the Mishna's connection of citation to redemption.
וָאֵרָ֗א אֶל־אַבְרָהָ֛ם אֶל־יִצְחָ֥ק וְאֶֽל־יַעֲקֹ֖ב בְּאֵ֣ל שַׁדָּ֑י וּשְׁמִ֣י יְהוָ֔ה לֹ֥א נוֹדַ֖עְתִּי לָהֶֽם׃
I appeared to Abraham, Isaac, and Jacob as El Shaddai, but I did not make Myself known to them by My name יהוה.
Aren't there plenty of examples of the Patriarchs hearing or invoking God's special four-letter name? R' Hirsch explains that "knowing" ("דעת") a name of God is something deeper than just being acquainted with it. It means fully understanding God's mode of interaction with the world that that name represents.
The four-letter name, in particular, refers to God in that he "מהוה חדשות" - "exerts His Will quite independently of existing conditions, even completely in spite of, or against them." Other names mentioned in this section, Elohim and El Shaddai, refer to God as we know Him most of the time, behind the scenes, working through rather than against the natural order.
At this point in Jewish history, the Israelites' lives were very much the product of God hiding, as it were, and allowing events to play out naturally. The more powerful nation had subjugated the less powerful one, evil men were freely acting as cruelly as their hearts desired, and there was no apparent hope of change. But now, God was promising, He was going to show them something brand new - His own emergence into the spotlight, shoving the natural order aside, and creating a new nation out of a hopeless crowd of slaves. The Israelites were about to get to know, intimately, a new name of God.
Ever since then, our improbable continuing existence has served as living testimony to God Who Innovates, and to His demand that humans do more with our lives than simply drift in nature's course.
The Exodus was an Earth-shattering moment of revelation of God's great Name, but still, R' Hirsch says, the world has yet to fully understand that Name. He surmises that we might not truly get there until "the end of all happenings of history," finding support in Isaiah 52:6, where, speaking of the ultimate Redemption, the God tells us through the prophet:
לָכֵ֛ן יֵדַ֥ע עַמִּ֖י שְׁמִ֑י לָכֵן֙ בַּיּ֣וֹם הַה֔וּא כִּֽי־אֲנִי־ה֥וּא הַֽמְדַבֵּ֖ר הִנֵּֽנִי׃
Assuredly, My people shall learn My name, Assuredly [they shall learn] on that day That I, the One who promised, Am now at hand.
It sounds like this verse explicitly identifies the ultimate Redemption with people finally and truly knowing God's Name: We're here to attribute Authorship of world history to God, and to a particular Name of God. When we all really understand that attribution, we'll be experiencing Geula!
Let's go back to Avot. I suggest that any time we give credit where credit is due, as Esther credited Mordechai for uncovering Bigtan and Teresh's assassination plot, we are exercising the attribution muscles we need for our overriding mission - acknowledging God as the Creator and Master of the world. Perhaps, then, the Tana is telling us that each time we do a proper citation, we are indeed bringing the final Redemption closer, for which a more immediate redemption of some other kind is a fitting "measure for measure" reward.