I would start from the mechanical output requirements (torque, travel, speed, positional accuracy) and work back to spec the motor/transmission, the electrical supply required, and the driver needed to control that. Depending on the answer you may be able to use an Arduino clone with integral motor drivers, or a UNO or similar with a standard driver shield, or any Arduino with a separate driver board.
slow acting variable 'cam' using a solenoid isn't a problem
Porsche Design were working on V.V.Tt using solenoids at least 10 yrs ago.
As far as I know the only engines running V.V.T. are big diesels
If your looking to experiment on internal combustion motor there is a lot of information but search terms vary a lot
Here's a link to the 'high side' http://www.autospeed.com/cms/article.html?&title=Camless-Engines&A=0910
and this is the one I couldn't originally remember the name http://www.sturmanindustries.com/