The Greek verb mnaomai means to remember, to recall, to think of, or to call to mind. It appears in the passive as 'be mindful of.' The verb is used of human memory and of God's 'remembering' — which in biblical idiom is never merely cognitive but always involves action on behalf of the one remembered.
When the Bible says God 'remembers,' it means he acts. God remembered Noah (Genesis 8:1) — and the waters receded. He remembered Abraham — and rescued Lot (Genesis 19:29). He remembered Rachel — and opened her womb (Genesis 30:22). The dying thief's 'Remember me when you come into your kingdom' (Luke 23:42) was not a request for nostalgia but for saving intervention. Jesus's response — 'today you will be with me in paradise' — shows what divine mnaomai looks like.