Origins
The word 'memory' entered English in the thirteenth century from Anglo-French 'memorie,' from Latin 'memoria' (the faculty of remembering, a recollection, a historical account), from the adjective 'memor' (mindful, remembering, unforgetting). The Latin 'memor' is built on PIE *(s)mer- (to remember, to be mindful of), closely related to — and possibly a variant of — the broader root *men- (to think). The initial 's' is a mobile s-prefix that appears in some branches and not others, which is why 'memory' has no s- but Sanskrit 'smárati' (he remembers) does.
The form 'memor' itself is striking: it appears to be a reduplicated form, where the root *mer- is partially repeated ('mem-or'), as if the word enacts its own meaning — remembering by saying the root twice. This kind of reduplication is an ancient Indo-European device for expressing intensity or repetition, and it gives 'memor' a phonological expressiveness that has survived into its descendants.
The Latin family from 'memor' and 'memoria' is vast. 'Memorandum' (a thing to be remembered) is the gerundive — a note written so that something will not be forgotten. 'Memorial' (a structure or event preserving memory) and 'commemorate' (to remember together, com- + memorare) serve the social function of collective remembrance. 'Memoir' (a personal historical account) entered English from French 'mémoire,' which preserves both the masculine sense (a written record) and the feminine sense (the faculty of memory) as distinct words. 'Immemorial' (beyond memory, from time immemorial) marks the boundary where memory fails.
French Influence
The verb 'remember' has a different but related path: from Old French 'remembrer,' from Late Latin 'rememorari' (to recall to mind again), from 're-' (again) + 'memorari' (to be mindful of). The English word thus contains a double emphasis on recall — 're-' (again) + 'memor' (mindful) — as if one must actively call back what the mind has stored.
Through the Germanic branch, PIE *(s)mer- (with the s-prefix) produced Proto-Germanic *murnan (to grieve, to be anxious), giving Old English 'murnan' and Modern English 'mourn.' The semantic connection is revealing: to mourn is to remember the dead, to be unable to stop being mindful of loss. Grief is memory that refuses to release its object. Greek 'mérimna' (care, anxiety, worry) similarly connects remembrance to distress — the things we cannot forget are the things that trouble us.
Sanskrit 'smárati' (he remembers), 'smṛti' (memory, tradition), and the important philosophical concept 'smṛti' (remembered tradition, as opposed to 'śruti,' directly heard revelation) all descend from the same root with the s-prefix intact. In Hindu philosophy, 'smṛti' texts are those transmitted through human memory — a vast category including the Mahabharata, the Ramayana, and the law codes — distinguished from the Vedas, which are 'śruti' (heard directly from the divine). The distinction between remembered knowledge and revealed knowledge is, in Sanskrit, a distinction built on this very root.
Figurative Development
The modern technological sense of 'memory' — computer memory, RAM, storage — extends the metaphor from human faculty to machine function, a transfer that Turing and von Neumann made deliberately: the machine has 'memory' because it stores and retrieves information, as the mind does.