I just went through this and went over all the options. I decided that for my lathe (a Delta Rockwell 11X36), which has a continuously variable Reeve's drive, that a rotary phase converter was the best option for me. I paid $150 for a used one with a 3hp 3-phase motor. You can make one yourself very easily. My lathe has a 1hp 3-phase motor and the 3hp rpc is more than adequate. An additional benefit of the rpc is that you can run multiple machines on one rpc, such as a lathe and mill, or even a 3-ph table saw. In fact, you can run several simultaneously, within limits of current and hp rating.
A VFD offers variable speed (which I didn't need) by varying the current frequency to the motor. It is more expensive than a rpc and it can only be used on one machine. Also, you can overheat your lathe motor with a VFD by operating it at speeds below which it is designed to operate, unless you install an auxiliary fan to cool it, although, by most accounts this is a minor worry unless you use your lathe long hours a day. Additionally, with my rpc, my lathe's speed ratings posted on the plaques are correct, while if I used a VFD I would have to install a tachometer to know precisely the speeds the lathe was turning. If your new lathe does not have a vari-drive, like mine does, and requires stopping the lathe to change speeds, a VFD may be the way to go.
For a rpc, choose a 3-phase motor with the same or greater hp rating as your lathe. It should be rated for the same voltage as your single-phase power source (mine was 220v), and the lower the rpm rating the better (1725 vs 3450), since the higher rpm is louder and it will be running continuously. You can simply wire it to your single-phase power source, then to a properly rated switch, and out to your motor. With this simple setup, you can start the rpc motor with a rope to get the rpc turning, after which it will run on the single-phase power, while re-configuring the power to 3-phase and passing it on to your lathe motor via the power switch. You can also add a pony motor (such as a washing machine motor) or in-line capacitors and a starter switch to make the rpc self-starting. Not hard, nor technically difficult, and there are plenty of sources of info and plans on the Internet.
Good luck.
