The question of the "correct" color for rigging line is complicated. It has no simple answer - even if we restrict the discussion to a particular period and nation - and in many cases the best answer seems to be that nobody's sure.
I've read more about the British navy of the eighteenth and early nineteenth centuries than any other. During that period, an Act of Parliament required that all rope delivered to the Royal Navy be soaked in Stockholm tar. I've never seen Stockholm tar, but I'm reliably informed that it's a rich, medium brown in color. That's probably the appropriate basic color for the rigging of an eighteenth-century British warship.
Standing rigging (i.e., the relatively heavy lines that held up the masts - including the shrouds, stays, etc.) seems to have been coated, after it was in place, with a concoction containing tar and lampblack, among other things. If it wasn't black, it was pretty close. It apparently was applied with a crude brush, and renewed on a fairly regular basis; logbooks of British ships include frequent references to "hands employed tarring the shrouds" (or, sometimes, "tarring the yards," or "tarring the bends" - meaning the wales, the heavy belts of planking on the exterior of the hull near the waterline). That's why so many modelers use black thread for standing rigging.
Just how the ratlines were treated is a matter of some guesswork. Logic suggests that coating them with that same tar-and-lampblack mixture would have made them awkwardly stiff and perhaps even slippery. On the other hand, it's difficult to imagine that all the ratlines would be unrove every time the shrouds were tarred. The "Modeler's Page" on the website of H.M.S. Victory says her ratlines are "lightly tarred." That makes as much sense as anything, I guess. I suspect the practice varied from ship to ship and, I suspect, from country to country.
The same goes for the deadeye lanyards. Logic suggests that tarring them would have been counterproductive; they had to be heaved through the holes in the deadeyes every time the tension on the shrouds was adjusted. The evidence of contemporary paintings and, later, photos suggests, however, that the lanyards and the shrouds usually (though probably not invariably) were the same color (i.e., black or close to it).
I'm not sure when the practice of tarring the standing rigging started. I once discussed the matter with an extremely knowledgeable scholar from the British National Maritime Museum, in Greenwich; he dismissed the whole idea of having more than one color of rigging line as "some sort of American modeler's convention." I'm pretty sure he was wrong about that - but I don't know whether "blackened" standing rigging appeared in the eighteenth, seventeenth, or sixteenth century - or earlier, for that matter.
Contemporary models are of little if any help here. Most of them have been rerigged in the twentieth century; if the original rigging does survive, it's usually undergone so much aging that it's tough to tell what the original color was. And one should be very careful in referring to restored vessels or modern replicas. They almost invariably use modern synthetic line, which may or may not look much like its historical counterparts. The rigging of the Victory, for instance, is polypropylene or something similar. And the replica H.M.S. Bounty, in the movie with Anthony Hopkins and Mel Gibson, advertised her twentieth-century origin with running rigging that was almost pure white.
Merchant ships during the sailing ship period, from what I can tell, seem in general to have followed the lead of warships, but probably weren't as consistent with the initial Stockholm tar treatment. Photographic evidence regarding nineteenth-century American clippers and whalers, for instance, seems to suggest that standing rigging was generally darker than running rigging. I'm inclined to think that nineteenth-century running rigging was usually about the color we still associate with well-used hemp rope - a dull, slightly brownish grey. For standing rigging I'd favor an extremely dark (almost black) grey with, maybe, a hint of brown.
As I've said more than once before, I wish somebody would undertake a really thorough study of the changing colors of rigging line through the centuries - and from nation to nation. (It probably isn't safe to assume that Spanish, Danish, Dutch, and English ships of any given period used the same colors of rigging line.) I'm not at all sure enough reliable sources exist to support such a study - and I'm not sure I'd want to read it cover-to-cover. (Sounds like a good cure for insomnia.) Until such a work appears, though, I think the generalizations I've just described are fairly reliable.
One other point (which I've also mentioned elsewhere in the Forum more than once): there are two Golden Rules when it comes to rigging sailing ship models. Rule #1: If in doubt as to size, err on the small side. Rule #2: If in doubt as to color, err on the dark side.
I'm afraid this post has, as so often happens in this Forum, wandered quite a bit beyond what the originator, Dad's Girl, intended. But this is a rather interesting topic. Dad's Girl - please bear in mind the all-time, number one Golden Rule in model building: It's great if you learn something from it, but what's most important is to have FUN.