Since you do not mention a regulator, I assume you are operating without one. If you have a rectifier, you may need to do some additional measurements.
Based on your latest measurements and the motor data:
A 12V, 3W motor has a coil resistance of 48 ohms.
Using the 470uF cap, you measured a dV of 2.8 volts.
With a full wave rectifier, dT is 1/2 cycle at 60 Hz or 0.00833 seconds.
Using I=CdV/dT = (470E-6)(2.8)/.00833 = 158mA
Using this as your motor current, you have to determine how much ripple you want. If you want to keep ripple to about 0.5 volts, use the same equation and solve for a new C.
C=IdT/dV = (.158A)(.00833sec)/0.5V = 2632E-6 or 2632 uF.
If you calculate with the rated current of 250 mA, the capacitance for 0.5 volt ripple will be
C=IdT/dV = .250(.00833)/0.5 = 3956 uF
Choose a 4700uF/25V aluminum cap with a ripple rating of 0.25amp or better rating. (Most are rated at over an amp.) Always use a 0.1uF cap at the output of the supply or near the motor to reduce the high frequency transients if there are any. You can use parallel capacitors of (4) 1000uF or (2) 2200 uF.
As more capacitance is added, the dV will decrease. The minimum voltage will become higher and the maximum will become slightly lower.
Note that without a 12 volt regulator, you will always have some significant ripple, based on the capacitor you choose.