And God remembered Noah...
God remembered Abraham...
God remembered His covenant with Abraham, with Isaac, and with Jacob...
It is not that God forgot, but that now it was time for God to act! To step in on behalf of His chosen people, His nation of Israel, or individual believers who have placed their faith in Him - act on their behalf, move in response to their heartfelt cries and prayers, and resolve their dilemma or relieve their distress.
Now it was time to remember Noah, abate the waters off the earth, bring Noah, his family, and all the animals safely out of the ark:
Genesis 8:1-3 And God remembered Noah, and every living thing, and all the cattle that was with him in the ark: and God made a wind to pass over the earth, and the waters asswaged; The fountains also of the deep and the windows of heaven were stopped, and the rain from heaven was restrained; And the waters returned from off the earth continually: and after the end of the hundred and fifty days the waters were abated.
Now it was time to remember Abraham's prayers and protect his nephew, Lot:
Genesis 19:29 And it came to pass, when God destroyed the cities of the plain, that God remembered Abraham, and sent Lot out of the midst of the overthrow, when he overthrew the cities in the which Lot dwelt.
Now it was time to remember certain women, and answer their prayers for children:
Genesis 30:22 And God remembered Rachel, and God hearkened to her, and opened her womb.
1 Samuel 1:19 And they rose up in the morning early, and worshipped before the LORD, and returned, and came to their house to Ramah: and Elkanah knew Hannah his wife; and the LORD remembered her.
Now it was time to remember His covenant, and to deliver His people from their Egyptian bondage, to act on their behalf and fulfill His promises toward them:
Exodus 2:24 And God heard their groaning, and God remembered his covenant with Abraham, with Isaac, and with Jacob. (See also 6:5)
Psalms 105:8, 42 He hath remembered his covenant for ever, the word which he commanded to a thousand generations... For he remembered his holy promise, and Abraham his servant. (See the context of the chapter and notice how God acted on behalf of Abraham's descendants, the nation of Israel.)
Psalms 106:43-45 Many times did he deliver them; but they provoked him with their counsel, and were brought low for their iniquity. Nevertheless he regarded their affliction, when he heard their cry: And he remembered for them his covenant, and repented according to the multitude of his mercies.
Now it was time to remember Israel and deliver them from their enemies:
Numbers 10:9 And if ye go to war in your land against the enemy that oppresseth you, then ye shall blow an alarm with the trumpets; and ye shall be remembered before the LORD your God, and ye shall be saved from your enemies.
Now it was time to remember the weakness of His chosen people, and to show them His mercy:
Psalms 78:38-39 But he, being full of compassion, forgave their iniquity, and destroyed them not: yea, many a time turned he his anger away, and did not stir up all his wrath. For he remembered that they were but flesh; a wind that passeth away, and cometh not again.
Psalms 136:23 Who remembered us in our low estate: for his mercy endureth for ever:
Luke 1:54 He hath holpen his servant Israel, in remembrance of his mercy;
Now it was time to remember the lost, and to remember those who were earnestly seeking the truth and the true God from their heart, and bring them the Gospel of Jesus Christ so they could receive it and be saved:
Psalms 98:3 He hath remembered his mercy and his truth toward the house of Israel: all the ends of the earth have seen the salvation of our God.
Acts 10:31 And said, Cornelius, thy prayer is heard, and thine alms are had in remembrance in the sight of God.
Now it was time to remember end-times Babylon and judge her for her sins:
Revelation 16:19 And the great city was divided into three parts, and the cities of the nations fell: and great Babylon came in remembrance before God, to give unto her the cup of the wine of the fierceness of his wrath.
Revelation 18:5 For her sins have reached unto heaven, and God hath remembered her iniquities.
On a contrary note: when God says He will not remember someone or some nation because of their sins, then the passage indicates an active opposition against them. See Lamentations 2:1; Ezekiel 3:20; 33:13.
For further consideration (a project for me to dig in a bit more as well): look up all the passages where God's people were told to remember, or where they remembered or brought something into remembrance. When a Biblical promise, precept, person, or past history was brought to mind, they acted on it by faith. When reminded of God's promises, they stepped out in faith and served Him. When reminded of His people, they prayed for them, and praised the Lord for the work He was doing in their lives. When reminded of God's past deliverances and faithfulness, they worshipped Him and gave Him the glory. When reminded of His commandments to them, they obeyed.
Their remembrance was active - and so should ours be! The Biblical example is acting on that remembrance, stepping out in faith, or in obedience, or in prayer - not just momentarily thinking about something and then dismissing it from mind once again.
John 14:26 But the Comforter, which is the Holy Ghost, whom the Father will send in my name, he shall teach you all things, and bring all things to your remembrance, whatsoever I have said unto you.
For more studies in Genesis (with Genesis as the springboard):