What does it mean when God Remembers?

We have many verses talking about God "remembering."
God does not change, so what do these verses mean?

וַיִּזְכֹּ֤ר אֱלֹקִים֙ אֶת־נֹ֔חַ וְאֵ֤ת כׇּל־הַֽחַיָּה֙ וְאֶת־כׇּל־הַבְּהֵמָ֔ה אֲשֶׁ֥ר אִתּ֖וֹ בַּתֵּבָ֑ה וַיַּעֲבֵ֨ר אֱלֹקִ֥ים ר֙וּחַ֙ עַל־הָאָ֔רֶץ וַיָּשֹׁ֖כּוּ הַמָּֽיִם׃

Genesis 8:1

God remembered Noah and all the beasts and all the cattle that were with him in the ark, and God caused a wind to blow across the earth, and the waters subsided.

בראשית כ״א:א׳

וַֽיקוק פָּקַ֥ד אֶת־שָׂרָ֖ה כַּאֲשֶׁ֣ר אָמָ֑ר וַיַּ֧עַשׂ יקוק לְשָׂרָ֖ה כַּאֲשֶׁ֥ר דִּבֵּֽר׃

Genesis 21:1

יקוק took note of Sarah as promised, and יקוק did for Sarah what had been announced.

בראשית ל׳:כ״ב

וַיִּזְכֹּ֥ר אֱלֹקִ֖ים אֶת־רָחֵ֑ל וַיִּשְׁמַ֤ע אֵלֶ֙יהָ֙ אֱלֹקִ֔ים וַיִּפְתַּ֖ח אֶת־רַחְמָֽהּ׃

Genesis 30:22

Now God remembered Rachel; God heeded her and opened her womb.

שמות ב׳:כ״ד

וַיִּשְׁמַ֥ע אֱלֹקִ֖ים אֶת־נַאֲקָתָ֑ם וַיִּזְכֹּ֤ר אֱלֹקִים֙ אֶת־בְּרִית֔וֹ אֶת־אַבְרָהָ֖ם אֶת־יִצְחָ֥ק וְאֶֽת־יַעֲקֹֽב׃

Exodus 2:24

God heard their moaning, and God remembered the covenant with Abraham and Isaac and Jacob.

שמות ב׳:כ״ה

וַיַּ֥רְא אֱלֹקִ֖ים אֶת־בְּנֵ֣י יִשְׂרָאֵ֑ל וַיֵּ֖דַע אֱלֹקִֽים׃ {ס}

Exodus 2:25

God looked upon the Israelites, and God took notice of them.

ויקרא כ״ו:מ״ב

וְזָכַרְתִּ֖י אֶת־בְּרִיתִ֣י יַעֲק֑וֹב וְאַף֩ אֶת־בְּרִיתִ֨י יִצְחָ֜ק וְאַ֨ף אֶת־בְּרִיתִ֧י אַבְרָהָ֛ם אֶזְכֹּ֖ר וְהָאָ֥רֶץ אֶזְכֹּֽר׃

Leviticus 26:42

Then will I remember My covenant with Jacob; I will remember also My covenant with Isaac, and also My covenant with Abraham; and I will remember the land.

ויקרא כ״ו:מ״ד

וְאַף־גַּם־זֹ֠את בִּֽהְיוֹתָ֞ם בְּאֶ֣רֶץ אֹֽיְבֵיהֶ֗ם לֹֽא־מְאַסְתִּ֤ים וְלֹֽא־גְעַלְתִּים֙ לְכַלֹּתָ֔ם לְהָפֵ֥ר בְּרִיתִ֖י אִתָּ֑ם כִּ֛י אֲנִ֥י יקוק אֱלֹהֵיהֶֽם׃

Leviticus 26:44

Yet, even then, when they are in the land of their enemies, I will not reject them or spurn them so as to destroy them, annulling My covenant with them: for I יקוק am their God.

ויקרא כ״ו:מ״ה

וְזָכַרְתִּ֥י לָהֶ֖ם בְּרִ֣ית רִאשֹׁנִ֑ים אֲשֶׁ֣ר הוֹצֵֽאתִי־אֹתָם֩ מֵאֶ֨רֶץ מִצְרַ֜יִם לְעֵינֵ֣י הַגּוֹיִ֗ם לִהְי֥וֹת לָהֶ֛ם לֵאלֹקִ֖ים אֲנִ֥י יקוק׃

Leviticus 26:45

I will remember in their favor the covenant with the ancients, whom I freed from the land of Egypt in the sight of the nations to be their God: I, יקוק.

שמואל א א׳:י״ט

וַיַּשְׁכִּ֣מוּ בַבֹּ֗קֶר וַיִּֽשְׁתַּחֲווּ֙ לִפְנֵ֣י יקוק וַיָּשֻׁ֛בוּ וַיָּבֹ֥אוּ אֶל־בֵּיתָ֖ם הָרָמָ֑תָה וַיֵּ֤דַע אֶלְקָנָה֙ אֶת־חַנָּ֣ה אִשְׁתּ֔וֹ וַֽיִּזְכְּרֶ֖הָ יקוק׃

I Samuel 1:19

Early next morning they bowed low before the LORD, and they went back home to Ramah.
Elkanah knew his wife Hannah and the LORD remembered her.

The below chapter from the Guide for the Perplexed shows Maimonides' method for understanding these verses.
We can apply the same methodology here.

מורה נבוכים, חלק א' מ״ה:א׳-ה׳

׳שמוע׳ – מלה משותפת, תהיה בענין שמע האזן, ותהיה בענין הקבול. אמנם בענין השמע ״לא ישמע על פיך״, ״והקול נשמע בית פרעה״ – וזה הרבה;

וכמהו ברבוי – גם כן ׳שמוע׳ בענין הקיבול ״ולא שמעו אל משה״, ״אם ישמעו ויעבודו״, ״ולכם הנשמע?״, ״ולא שמעו אל משה״, ״אם ישמעו ויעבודו״, ״ולכם הנשמע?״, ״ולא ישמע את דבריך״.

ויהיה בענין הידיעה ״גוי אשר לא תשמע לשונו״ – פרושו: לא תדע דברו.

וכל מלת ׳שמיעה׳ שבאה בחק האלוק – אם יראה מפשוטו של כתוב שהוא מהענין הראשון הוא ספור על ההשגה והוא מהענין השלישי ״וישמע יי״, ״בשמעו את תלונותיכם״ – כל זה השגת מדע;

ואם נראה מפשוטו של כתוב שהוא מן הענין השני הוא ספור על ענותו ית׳ לצעקת הצועק למלא שאלתו או לא יענה לצעקתו ולא ימלא שאלתו ״שמוע אשמע צעקתו״, ״ושמעתי כי חנון אני״, ״הטה יקוק אזנך ושמע״, ״ולא שמע יקוק בקולכם ולא האזין אליכם״, ״גם כי תרבו תפלה אינני שומע״, ״כי אינני שומע אותך״ – וזה הרבה.

והנה יבואך באלו ההשאלות והדמיונות מה שירוה צמאך ויבאר ספקך ויתבארו לך עניניהם כולם עד שלא תספוק בדבר מהם.

Guide for the Perplexed, Part 1 45:1-5

Shama‘ is used homonymously. It signifies “to hear,” and also “to obey.” As regards the first signification, comp. “Neither let it be heard out of thy mouth” (Exod. 23:13); “And the fame thereof was heard in Pharaoh’s house” (Gen. 45:26). Instances of this kind are numerous.

Equally frequent are the instances of this verb being used in the sense of “to obey”: “And they hearkened (shama‘) not unto Moses” (Exod. 6:9). “If they obey (yishme‘ü) and serve him (Job 36:11); “Shall we then hearken (nishma‘) unto you” (Neh. 13:27); “Whosoever will not hearken (yishma‘) unto thy words” (Josh. 1:18).

The verb also signifies “to know” (“to understand”), comp. “A nation whose tongue, i.e., its language, thou shalt not understand” (tishma‘) (Deut. 28:49).

The verb shama‘, used in reference to God, must be taken in the sense of perceiving, which is part of the third signification, whenever, according to the literal interpretation of the passage, it appears to have the first meaning: comp. “And the Lord heard it” (Num. 11:1); “For that He heareth your murmurings” (Exod. 16:7). In all such passages mental perception is meant.

When, however, according to the literal interpretation the verb appears to have the second signification, it implies that God responded to the prayer of man and fulfilled his wish, or did not respond and did not fulfil his wish: “I will surely hear his cry” (Exod. 22:23); “I will hear, for I am gracious” (ib. 27); “Bow down thine ear, and hear” (2 Kings 19:16); “But the Lord would not hearken to your voice, nor give ear unto you” (Deut. 1:45); “Yea, when ye make many prayers, I will not hear” (Isa. 1:15); “For I will not hear thee” (Jer. 7:16). There are many instances in which shama‘ has this sense.

Remarks will now be presented to you on these metaphors and similes, which will quench your thirst, and explain to you all their meanings without leaving a doubt.

We can apply the above principal to understand what it means when God is said to "remember"
It means that that there is a change in the observable state in the world.
It means that a covenant that has always been present has now been realized.

Note how the below Zohars link the idea of God remembering to a change in the state of the world, not a change in God's mind.

ספר הזהר, נח כ״ט:רט״ו

(ונענשו), כֵּיוָן דְּאִסְתַּתַּר וְאִשְׁתָּהֵי תַּמָּן כָּל הַהוּא זִמְנָא. לְבָתַר וַיִּזְכֹּר אֱלֹקִים אֶת נֹחַ. אָמַר רַבִּי שִׁמְעוֹן (ויזכור אלקים את נח) תָּא חֲזֵי בְּשַׁעֲתָא דְּדִינָא אִתְעֲבִיד לָא כְּתִיב בֵּיהּ זְכִירָה. כֵּיוָן דְּאִתְעֲבִיד (ביה) דִּינָא וְאִתְאֲבִידוּ חַיָּיבֵי עָלְמָא, כְּדֵין כְּתִיב בֵּיהּ זְכִירָה. דְּהָא כַּד דִּינָא שַׁרְיָא בְּעָלְמָא אִתְחַבְּרוּתָא לָא אִשְׁתְּכַח. וּמְחַבְּלָא שַׁרְיָא עַל עָלְמָא.

Zohar, Noach 29:215

What does it say afterward? It says, "And Elohim remembered Noah." So Rabbi Shimon said, Come and behold: While Judgment is executed, there is no remembrance. But after Judgment has been executed and the wicked have been removed from the world, then the word "remember" is mentioned, because when Judgment prevails in the world, there is no (supernal) mating above and the Angel of Destruction rages throughout the world.

ספר הזהר, נח כ״ט:רט״ז

כֵּיוָן דְּאִתְעֲבַר דִּינָא וְאִשְׁתַּכִּיךְ רוּגְזָא, תָּב כֹּלָּא לְאַתְרֵיהּ. וּבְגִין כָּךְ כְּתִיב הָכָא וַיִּזְכֹּר אֱלקִים אֶת נֹחַ. דְּבֵיהּ שַׁרְיָא זָכוֹר. דְּנֹחַ אִישׁ צַדִּיק כְּתִיב בֵּיהּ.

Zohar, Noach 29:216

But after Judgment is passed and is completed, and the wrath is appeased, everything returns to its proper order. This is why the text reads, "And Elohim remembered Noah." Because the term "remember" applies to Noah, as indicated by the words, "Noah was a righteous man."

זוהר חדש, פרשת נח כ׳

ר''י אֲמַר, נֹחַ כַּד הֲוָה בְּתֵיבוּתָא, הֲוָה דָחֵיל דְּלָא יִדְכַּר לֵיהּ קוּדְשָׁא בְּרִיךְ הוּא לְעָלְמִין, וְכֵיוָן דְּאִתְעֲבָרוּ חוֹבֵי עָלְמָא, כְּדֵין מַה כְּתִיב וַיִּזְכֹּר אֱלֹקִים אֶת נֹחַ.

Zohar Chadash, Noach 20

Rabbi Yehuda says: Noach, when he was in the Ark, he was afraid that the Holy One, blessed be he, would not rembember him ever. And then the evil people of the world were gone, what is written? And God remembered Noah.

There is a connection between the three concepts of:
1. Covenant (brit)
2. Rightiousness (tzadik)
3. Remembering (zachor)
Remembering of God is remembering a covenant - a bond that transcends time and is not always seen. Tzadik is someone who acts in line with a bond that will be remembered.

ספר הזהר, נח כ׳:קנ״ה

וַהֲקִמֹתִי אֶת בְּרִיךְ אִתָּךְ, לִהְיוֹת אוֹת בְּרִיתִי בָּעוֹלָם, וְאַחַר כָּךְ וּבָאתָ אֶל הַתֵּבָה. שֶׁאִלְמָלֵא אֵינוֹ צַדִּיק (ברית), לֹא יִכָּנֵס לַתֵּבָה. שֶׁהֲרֵי לֹא מִתְחַבֵּר לַתֵּבָה, רַק צַדִּיק. וְלָכֵן וּבָאתָ אֶל הַתֵּבָה, וַהֲרֵי זֶה נִתְבָּאֵר.

Zohar, Noach 20:155

"But with you will I establish My Covenant," - you shall be My Covenant in the world. And after that, "you shall come into the ark." Because had he not been righteous, he would not have entered the ark, as only a righteous man may be connected with the ark. That is why it is written, "you shall come into the ark,"

Additional Questions:

What is the connection between Tzadik, rightiousness, and Covenant?
What does it meam when we call God a "Tzadik"?
What does the Ark represent?
What does the concept of Rosh HaShana have to do with the New Moon?