Forgot to say in the notes, but Nikki doesn't literally say "engrave". She uses the general verb 'turn into/become' (思い出になる). It's just that the overall dialogue is more vague and sorta centered on the word memory instead (思い出) which is also in 'being able to remember' (思い出せる). 'Literally' they also say it's "a song with(?) a memory" instead, which in English is "a song that brings back memories". With this general shift between languages in mind, I thought it'd be better to have other specific expressions rather than keeping it vague.
So "engrave on someone's memory" (or etch, and I think it's more common to say 'engrave' on your heart/mind, and 'etch' into your memory). Either way, etching and engraving refer to cutting a pattern on the surface of a hard object. Same as you do with music records, so I think it's more fitting for a song. But actually, when it comes to records you usually say "cut into" instead, since nowadays etching/engraving is also used for a process where you print a special, decorative design on the disc.