I find from a web search that novel is the humorous novel, "Three Men in a Boat (To Say Nothing of the Dog)" by Jerome K. Jerome. I've read the book more than once, and I love it, but I didn't remember the phrase. It's always helpful to mention the exact source when you ask a question about a quotation. Two points to note is that this is British English, and that it was written in 1889. So, it not only is British English, but it might be old-fashioned or out-of-date British English. I hope a British native speaker will answer your question. As a US speaker, if I heard this, I would assume the same thing as the other people who've answered. Counting backwards: last Friday, the Friday before last, the Friday two weeks before, then the Friday three weeks before. If it is April 26th, that would be March 30th. But I could easily be wrong by a week. In real life, if it were important to name a date like that, I would never use an expression like that; and if someone used it, I would say "I'm sorry, you'd better tell me the actual date, did you mean March 30th?" I think the writer is making a joke by pretending that the person can remember it so accurately. The topic is that people lie about seasickness. The narrator says that most people get seasick but most people say that they don't. "It is a curious fact, but nobody ever is sea-sick—on land." The narrator says he saw someone being seasick on the Yarmouth boat, then, "three weeks later," heard the man telling people that he was never seasick. The narrator decides to challenge the man. He says he saw the man not only being seasick but so seasick that "he wanted to be thrown overboard" (wanted to die, it was so bad). The man pretends not to remember. The narrator retorts by naming the exact day when it happened. It reminds me of a joke in which someone says to someone else, "Aren't you ever wrong?" and the person replies, "Absolutely. The last time I was wrong was on April 23rd, 1993 when I said..."

It is an odd way to say it, but it means the Friday three weeks prior to the Friday that most recently past (which is "last Friday"). More common ways to say it: three weeks ago last Friday or three weeks ago this past Friday.

It doesn't make sense but "three weeks [ago] last Friday" makes perfect sense.