Throughout my time playing New Leaf, a lot of my villagers have been asking if the rumours of me time-travelling were true. Is there any advantages or disadvantages associated with any of the options?
Probably based on the mechanic of rumors in the game. For instance, some of my villagers will say if the rumors of me being an "aspiring designer" or a "pro listener" are true. It changes depending on what you've been doing in-game recently. "Stump maker" is one, "bell pincher", etc. And yeah, "time-traveler" is one of those things.
It's just one of the normal things villagers like to say about you. No real ups or downs to it unless you want a specific title.
It's nothing to worry about, just villager talk. c: Like PotatoMuffin said, rumors are just based on the thing you did the most last. A few days ago I also had the "time traveler" rumor because I consecutively TT'd a few days ahead to move someone out, then afterwards I talked to all my villagers a bunch of times, and the "I heard you're quite the time traveler!" rumors quickly changed to "I heard you're quite the pro-listener!" soon after. No real consequences.