What you are looking for is a Lazy Susan (or, sometimes, turntable). They vary between little plastic things you can pick up at a housewares store that will hold a couple of kilograms at most to very expensive rigs that can turn a diesel locomotive with ease. The chances are pretty good that you'd want something in the low-middle end of that range that can be a part of your studio for a while.
While it may be more money than you had hoped to spend, I'd suggest investing in about a 30-inch set of aluminum bearings with about a half-ton rating. It's not that you're going to want to be lifting a half-ton of anything onto a table over and over again, but you want the stability, repeatability and longevity that will make the setup worth your while. A good set of bearings along with a sheet of 3/4" MDF (half above, half below) will give you a solid, stable platform that will last for years of product shooting. Even if you never shoot another spinner again, it's often easier to adjust the table upon which the product is sitting than it is to move the camera, the lights, the reflectors, the scrims, the flags,...
It's not too difficult, with the right tools and a bit of skill (or a hired cabinetmaker) to create a rectangular table with a revolver circular bit in the middle and just the smallest gap. It's easy to heal or clone out the gap — it's even something that would have been relatively easy in the days of film. With a white melamine or laminate surface, you can even extend it into a seamless sweep.
How many images you'll need will depend on how quickly and smoothly the object is supposed to spin. 15 degrees between images ought to do it, but I'd check for a "tighter" brief first.
You're a photographer. You make the "big bucks" with your camera. You can't charge nearly as much for post-production as you can for shooting, and if you bring your post-production hourlies into the mix at all, you'll find the client won't be willing to pay very much for it. You'll need to consider PP for your own purposes, but you're charging for the product shot. Every product or every different shot of the same product (not a rotate series, but, say, a different product orientation, accessories, props, etc.) is a separate line item on the invoice, as is the session base if you have to travel, any additional crew or talent you need, etc. But the shot (or rotate series) is a deliverable, and it doesn't matter how much behind-the-scenes has to happen for you to deliver it — post-processing isn't a separate charge unless the client wants a change.
That means two things: get fast at the things you have to do, and do whatever you can at shooting time to eliminate as much post-processing as possible. It's likely, for instance, that building a decent Lazy Susan shooting table will set you back $300 or more in materials and a full day in time, but it will pay for itself in time saved on a single 24-image rotation sequence compared to putting the object on a chair you have to edit out of every shot. Sometimes you have to spend to make.
