I've just been through this myself.
Converting 15MSH from tiller to remote?
The thing you have to be very careful about is just which control box you order, assuming that you will need one. They come in several flavors and finding the correct part number is very difficult, especially if you want a stop button on it.
You need to know if the control will be mounted on the Stbd, or Port side. You also need to know if it is to be configured as a "push" to open the throttle or "pull" to open the throttle. If you order the wrong side, you are pretty much screwed. If you order the wrong "push or pull" you are almost screwed. You can convert one to the other but it means some grinding, drilling and tapping of components to make them work, not easy, but certainly not impossible.
Determining if you have a "push" or "pull" to open throttle while you still have the tiller on is very hard to determine for sure. The throttle quadrant that moves the throttle turns clockwise to open as viewed from the right side of the engine. The cables pull from both "sides" of the quadrant. Is it push or pull?
The only way to know for sure is to have the conversion kit installed. The conversion kit has a lever that the cable connects to. It mounts so that the connecting point is below the shaft, which means the cable has to push to move the quadrant clockwise. At least, this is how it's done on my engine. If you look closely at the parts diagram for the conversion kit it shows this lever in that position, but determining relative positions form a parts diagram can be risky.
You will also need the steering hook and attaching bolt assembly as well as a longer tilt tube (bracket bolt) as the one on the tiller engine is not long enough to attach the steering cable to properly.
Depending on where you shop you will be looking at between $650 and $1000 for the parts you will need. The kits come with instructions which are not bad, but could be better.
Hope this helps.