When we are introduced to Avraham Avinu in Lech Lecha, Hashem promised him וְאֶֽעֶשְׂךָ֙ לְג֣וֹי גָּד֔וֹל. That promise would be realized through experiencing the Egyptian bondage. When Yaakov is on his way down to Egypt (46:3), Hashem appeared and told him אל תירא מרדה מצרימה כי לגוי גדול אשימך שם. What makes a גוי גדול? The Torah tells us later in Devarim (4:8) ומי גוי גדול אשר לו חקים ומשפטים צדיקם ככל התורה הזאת. A ג֣וֹי גָּד֔וֹל means a nation that stands out in its spiritual, religious, moral commitment. Therefore, the Shabbos when Bn"Y is given its first mitzvah, to take a korban pesach, is called "Shabbos haGadol." This is our inauguration into becoming a people dedicated to mitzvos.
The Sefas Emes (5652) develops this theme further in a beautiful piece, but I would like to suggest a different spin. We find that word גָּד֔וֹל used elsewhere: פְּנ֣וּ׀ וּסְע֣וּ לָכֶ֗ם וּבֹ֨אוּ הַ֥ר הָֽאֱמֹרִי֮ וְאֶל־כׇּל־שְׁכֵנָיו֒ בָּעֲרָבָ֥ה בָהָ֛ר וּבַשְּׁפֵלָ֥ה וּבַנֶּ֖גֶב וּבְח֣וֹף הַיָּ֑ם אֶ֤רֶץ הַֽכְּנַעֲנִי֙ וְהַלְּבָנ֔וֹן עַד־הַנָּהָ֥ר הַגָּדֹ֖ל נְהַר־פְּרָֽת (Devarim 1:7) Why is Pras called הַנָּהָ֥ר הַגָּדֹ֖ל? Rashi there explains that it is because Pras is mentioned in context and in connection with Eretz Yisrael. מפני שנזכר עם ארץ ישראל, קורהו גדול.
The context of the promise of אֶֽעֶשְׂךָ֙ לְג֣וֹי גָּד֔וֹל is Avraham being told in Lech Lecha to leave his homeland and go to Canaan/Eretz Yisrael. It's developing that that connection to Eretz Yisrael which is what will make him into a ג֣וֹי גָּד֔וֹל because, as Rashi tells us, gadlus comes through connecting with our homeland.
We find this connection elsewhere. Simple pshat in the pasuk in Yehoshua (14:15) וְשֵׁ֨ם חֶבְר֤וֹן לְפָנִים֙ קִרְיַ֣ת אַרְבַּ֔ע הָאָדָ֧ם הַגָּד֛וֹל בָּעֲנָקִ֖ים ה֑וּא וְהָאָ֥רֶץ שָֽׁקְטָ֖ה מִמִּלְחָמָֽה is that there was a man named "Arba" who was a big giant, הָאָדָ֧ם הַגָּד֛וֹל בָּעֲנָקִ֖ים, and lived in Chevron. However, if you look in the Midrash (e.g. see Yalkut Shimoni) you will find that Chazal associate הָאָדָ֧ם הַגָּד֛וֹל בָּעֲנָקִ֖ים with Avraham Avinu. Aside from the derash reasons given, I would suggest that Avraham deserves this title particularly in connection with Chevron because that is where Avraham made the first purchase of land in Eretz Yisrael.
Perhaps this is also the meaning of the promise made in bris bein ha'besarm, where Avraham is told (15:14) אַחֲרֵי־כֵ֥ן יֵצְא֖וּ בִּרְכֻ֥שׁ גָּדֽוֹל. The רְכֻ֥שׁ גָּדֽוֹל that we were supposed to look forward to is the return to Eretz Yisrael.
After introducing the halachos of korban pesach, the Torah tells us וְהָיָ֞ה כִּֽי־תָבֹ֣אוּ אֶל־הָאָ֗רֶץ אֲשֶׁ֨ר יִתֵּ֧ן ה׳ לָכֶ֖ם כַּאֲשֶׁ֣ר דִּבֵּ֑ר וּשְׁמַרְתֶּ֖ם אֶת־הָעֲבֹדָ֥ה הַזֹּֽאת. The parsha is reiterating the promise of return to Eretz Yisrael, the land of the Avos, and telling us that it is there that we will continue our future avodah. In fact, according to the Targum Yonasan, the "kanfei nesharim" that the Torah later speaks of refers to Bn"Y magically being transported on the seder night in Mitzrayim to Yerushalayim to offer their korban pesach there.
My suggestion is that Shabbos haGadol is given that name because the command to offer korban pesach anticipates our return to Eretz Yisrael, the renewal the avodah of the Avos in out homeland.