First of all, it was not "tiddleywinks" originally, which is how we always said it in my family. Somehow the third D got lost over the years.
Merriam-Webster's site backs the original spelling up; describing it as "a game whose object is to snap small disks from a flat surface into a small container."
And here it is!
We actually did play some Tiddledywinks when I was a kid; it was something my parents remembered from their own childhood (not in 1890, no). As you can see by the illustration, the game is similar in its way to Quarters, only with a more sophisticated and trickier delivery method and without the vomiting.
I don't suppose I need to tell you that there are entire Web sites devoted to Tiddledywinks, including that of the North American Tiddlywinks Association, from whence the above image comes. It gives you the rules and other helpful information, especially good if you want to set up your own league or something. You will notice that the site does spell it without that third D, though---apparently the rampant popularity of the game led to many imitators and I suspect unlicensed copycats, and you always get knockoff spellings in cases like that. A long history of the game can be found at the site as well, a history that explains another aspect of the strange case: How did Tiddledywinks come to be synonymous with trivial pursuits? (As it was put in my family, "What are ya doing, playing tiddlywinks?" or "Well, I ain't playing tiddlywinks!")
"Tiddlywinks was an adult craze in the 1890s, then fell into 'disrepute' as a simpleminded children’s game," said the author, and that's where my folks picked it up. Whether my parents did play the game much in their own childhoods, I'm not sure; I know my dad was more a pitch penny and stoopball kid on the city streets. Mom always loved the indoor games, though.
One of the comments on the Merriam-Webster page notes that in his childhood they used the term to mean something of little value, and gives the example "What you are doing is not worth tiddlywinks." He is surprised as I to find this meaning not listed on the dictionary page.
Personally, I suspect the English Tiddlywinks Association got to the dictionary editors. It wouldn't take too much. Squop a few strategic winks and the rest will fall in line.