'Remember' traces to PIE *(s)mer- — not just thought, but anxious, caring attention that refuses to let go.
To have in or be able to bring to one's mind an awareness of something previously known or experienced.
From Middle English 'remembren,' borrowed from Old French 'remembrer' (to remember, recall to mind), from Late Latin 'rememorārī' meaning 'to remember again, to call to mind,' a compound of Latin 're-' (again, back) and 'memorārī' (to be mindful of), from 'memor' (mindful, remembering), from PIE root *mer- or *(s)mer- (to think, to remember, to be anxious about). The Latin root 'memor' also produced 'memory,' 'memoir,' 'memorial,' 'memorize,' and 'commemorate.' The English word displaced the native Old English 'gemunan'
Though 'remember' looks like it should be related to 'member' (as in body part) and 'dismember,' the connection is coincidental in English but real in folk etymology. However, 'dismember' actually IS from a different Latin root: 'membrum' (limb). The 're-member' pun — to 're-member' a dismembered thing by putting its members back together — is a happy accident of English spelling, not genuine etymology.