remembrance: holding a person in mind and naming them, above all before God in prayer
All seven times this word appears in the New Testament it appears in a letter of Paul, and almost always in the thanksgiving he puts at the front of the letter. He tells a church or a friend that he keeps them in mind and says their names to God. The word covers both halves of that at once: the inward act of not forgetting a person, and the outward act of speaking their name out loud in prayer.
Four of the seven use a set phrase that letter writers of that era used constantly, literally "to make remembrance of" someone. In ordinary correspondence it was a courtesy, a way of assuring a friend that you named them before the gods. Paul takes the courtesy and fills it with real content: he is not being polite, he is reporting a habit he says he keeps without ever stopping.
The remaining three turn the word in two directions. In Philippians and 2 Timothy it is Paul's own steady remembering, tied straight to his praying. In 1 Thessalonians 3:6 the direction flips: the Thessalonians are the ones doing the remembering, and their memory of Paul is called a good one, which is how friends who miss each other talk. Nearby words divide the work differently. μνήμη is memory as a faculty or a lasting record, and ἀνάμνησις is a deliberate act of calling something back to mind, as at the Lord's Supper.