If you have it in your budget, I would definitely suggest purchasing a seat for your son. Depending on the airline, sometimes they will give you a 1/2 price ticket for a child under two. If you are booking through a travel agent, they may be able to get you that deal. When I flew alone with my oldest son when he was six months old I got a 1/2 price ticket on United, but last month, my husband and I flew with both our kids (now 3YO and 18 months) and that was not an option on Jet Blue, so we diceded to have my 18MO be a lap rider, and that was tough at that age. I was a little cramped with one on my lap and anticipate the flight would have been tremendously more comfortable for both of us if he had his own seat. Even if you don't purchase a seat, sometimes the gate agent will give you an extra seat if the plane is not full, but ours was, so don't count on that. Check with the airline about car seat rules as well. Sometimes under two, they MAKE you have a car seat on the plane, which can be a pain, so that might factor into your decision as well.
Our flight was five hours, and I have to say the boys did remarkably well. I purchased some inexpensive toys and coloring books that they had not seen before, and brought mini tubs of play-dough to keep them busy (I was worried they play dough might not make it through security, but it did).
It is a tough decision whether to schedule layovers and make the traveling take longer, or do a non-stop and just get it over with. We chose a non-stop so we could just get on the plane, and settle in for the long haul, rather than having to pack up, get off, wait, get on another plane, get settled again, and so on. So, even though 15 hours on a plane may be seem like a challnege, I would choose to tough it out on a nonstop, instead of having to takeoff/land, get on/off, and wait multiple times. I found that even in the airport we had to keep them strappend in strollers, as it wasn't really an ideal place for them to run around and play, so it wouldn't seem to me like it provide much of a break anyway, especially if you are running late for your connecting flicght (anything can happen) you just end up making a mad dash to get on another plane and get no break at all.
Either way, don't stress. It's not going to be nearly as bad as you anticipate, and the flight is not going to be what you remember anyway, as you will create so many wonderful memories for your family with the expereince. Good luck and have FUN!