How old?
With a delay like that, I always think of a fuel delivery problem first. A problem with things like plugs or injectors generally doesn't wait for 3 seconds to show itself.
Does it have a primer bulb, and is the bulb getting sucked flat?
I'd always change the fuel filter first, especially if you haven't replaced it yet, and double especially if it's a water separating fuel filter. It's amazing how quickly and easily a fuel filter gets clogged on a boat of any significant age. Buy two, so you always have a spare on hand when you actually need it, instead of on Sunday when everyone is closed or no one stocks it. Also, buy OEM filters because they're still a relatively cheap part, and the generic aftermarket filters (Sierra, etc.) are often only available in larger micron ratings.
There are some other easy things too, like not waiting for the ECU to fully boot on some outboards until you hear the beep (like Evinrude, not sure about Mercury stuff). If this is not done on an Evinrude ETEC for example, the min throttle position adapt is not done properly and can cause all kinds of stumbling and hesitation problems (the fix is to turn off, and re-start properly the next time).
If you haven't had the boat long, you might be overlooking something simple from lack of familiarity.
