Recommendation for an inexpensive electric motor to drive a mechanical speedometer
If you want this to be an open loop solution (ie just fit a motor and be able to designate the peed without complexity) then a stepper motor would be ideal. 

Stepper motors are not inherently slow as some may claim, most small Nema 8 , 11 and 17 steppers will accurately follow up to 8000 pulse per second full step. With a 1.8deg, 200 steps per rev motor this gives you 40 revs/second or 2400 rpm. 

You could easily source the components for your project from those cheaply available for small DIY 3D printers (these will typically be Nema 17). 

While many of the steppers on Ebay will have poor datasheets, here is one from a larger manufacturer that shows the step rate and torque curves that show it being viable out to 38 rps. This is a smaller Nema 11 sized motor. 

A lot will depend on how you want to engineer your solution. do you want it to attach directly to the back of the speedo (very low torque requirements) or under the hood and attached to what was the gearbox end of the speedo cable (much higher torque).
Either way, the physical configuration of the stepper motor allows very robust engineering practices, so making an adapter into the back of the speedo seems doable. 

While slightly harder to drive, you could consider a 3 phase BLDC motor with excellent torque.  The typical DC motor with brushes I would not recommend since brush wearout will be a problem. Most of these small DC motors are only rated for a few hundred hours of operation. 

Update: there are certainly ways to run a stepper motor without any noise if you are going to mount it directly to the speedo. 
The Trinamic series of drivers (2208) are quite cheap and are essentially silent in operation (by using sine wave chopper drive). However you will have difficulty reaching 30+ rps with something like an Arduino.  

Remember that the stepper motors will provide much more torque than you need when directly connected to the speedo head, so you could 'gear up' using a belt drive, again these parts are readily available for DIY 3D printers, so cost is relatively low. You may be able to gear-up by 3:1 which would suit the Trinamic driver.