I feel like there's two parts to this question:
1. Logical punctuation vs. typesetter's punctuation
2. How to end a narrative quote
For 2, every single thing I've seen says that when you're quoting, if the original ends in a period, you just turn it into a comma and don't have anything extra after the quote. For 1, it's debatable and could go either way depending on your preference. So, to my knowledge, the two best ways to do it would be either "There is a thread about random stuff; it's called 'Random Stuff Part III'," said silver. with logical punctuation, or "There is a thread about random stuff; it's called 'Random Stuff Part III,'" said silver. with typesetter's punctuation.