Left speech radical - the indexing component, contracted from 言. Marks 记 as a verbal act: to speak something into memory, to record or note it down. Same family as 说 (speak), 话 (talk), 谢 (thank), 课 (lesson).
Right 己 supplies the sound - jǐ drifted to jì with tone shift. Same phonetic appears in 纪 (record), 忌 (taboo), 起 (rise). The 'self' meaning faintly resonates: to remember is to inscribe something on the self.
