It's a bit late at present, but I think the museum might accept the broken clock, especially if it was restored, but perhaps even not. An antique from around the creation of New Hometown! That sounds just like the sort of thing they'd display. But of course, we're already here at the bridge to get a super old rock.
...although... if the bridge is longer than it used to be, it may have been extended over the years at some point, perhaps as the space it needed to span got wider. That would mean some rocks might not be appreciably old enough. (Technically, almost any rock is probably ancient, though... it'd hard to tell what Ran might consider acceptable when you consider something as mundane as a rock.)
