Yes, keepsake occurred to me after the post. And maybe memento. Although new things can be keepsakes & mementos too. "Family treasures" is a good expression; that may come closest to what I'm thinking of although it implies multiple people.

Related adjectives, but these also don't necessarily imply age: Cherished, treasured.
Related nouns: nostalgia, sentimentality.

I still keep thinking there must be a noun (possibly from another language) for a "quality an object obtains in increasing quantity with age and familiarity (and fondness)".
It's not an intrinsic quality of an object, but the perceived quality / feeling it invokes in someone.
Sort of like "value", but that word seems so vague.

Ah, now I may have it, although in two words: "Nostalgic value". Or "Sentimental value".