The Bible was not originally written in a modern language. The Old Testament was written in Hebrew and the New Testament was written in Greek. Most of the references to God "remembering" something are in the Old Testament. So we really have to look to the Hebrew to deduce the meaning of the phrase "God remembered."
When the Bible says God remembered something, what does that mean?
Biblical Hebrew includes only about 8,000 words arising from only about 2,100 root words as compared to the over 400,000 words we have in English. Because of the smaller vocabulary, words in Hebrew often encompass multiple meanings and a range of ideas. In particular, Hebrew verbs that English speakers consider to be merely mental processes actually include the physical outcome or expected activity of that mental process. For instance, the verb shema translates "to hear," but actually means "to hear, believe, and obey/act upon what was heard."
When the Bible says God "remembered," the original Hebrew verb is zakar. Zakar does mean "to remember," but it also means "to bring someone to mind and then act upon that person's behalf." The Hebrew idea of remembering always includes acting on behalf of the one brought to mind.
The biblical examples of "remember" including action on the other's behalf are numerous. When God "remembered Noah… [He] made a wind blow over the earth, and the waters subsided" (Genesis 8:1). God first turned His attention to Noah and then acted on Noah's behalf. When "God remembered Rachel,… God listened to her and opened her womb" (Genesis 30:22). Similarly when Joseph prophesied that Pharaoh's cupbearer would be restored from prison to his previous position, he asked that the cupbearer "mention me to Pharaoh, and so get me out of this house" (Genesis 40:14). "Mention" in this verse is actually the Hebrew zakar, as in "remember me to Pharaoh" with the stated goal of getting Joseph out of prison.
When the Bible says that God "remembers" something or someone, it means He turned His attention to someone and acted upon their behalf. That is why the psalmist in Psalm 106:4 calls out, "Remember me, O LORD, when you show favor to your people; help me when you save them." The cry is not that God has forgotten that the psalmist exists, rather it is a cry for God to turn His attention toward the psalmist and rescue him from his situation.
Similarly, in Jeremiah 31:34, when God describes the new covenant, He says "For I will forgive their iniquity, and I will remember their sin no more." Here God is promising not necessarily to forget sin, but rather to not act in accordance to what their sin deserves. He is "not remembering" in the very Hebrew sense of not acting upon that sin. In fact, Mary makes the connection of God's "remembering" being a very physical act when she equates the living Messiah in her womb as God's "remembrance of his mercy" (Luke 1:54).
We can be so thankful that God does remember us, has turned His attention to us, and has acted upon our behalf by sending His son so He can "not remember against us our former iniquities" (Psalm 79:8). What a blessing!
What does it mean that God forgets our sin? Does He forget?
Is God omniscient? What does it mean to be omniscient?
The attributes of God, what are they?
How does the Bible use anthropomorphism to talk about God?
How can doing a word study be helpful in understanding the Bible?
Truth about God