How to analyitically optimize an RLC "circuit" to reduce overshoot?
With your physical measurements, you have two knowns, the observed frequency, which will be close to the natural frequency, and the rate at which the oscillation decays, related to the Q. The easiest way to record the latter is the reduction in peak amplitude for each successive swing.

Although you think you have three unknowns, R, L and C, you only really need two at this stage, because the value of any one of these is unknown to a scaling factor, related to the impedance of the circuit. Your two unknowns can be any two ratios, conveniently L/R and C/R. You could invert the equations for frequency and damping and solve for the two ratios, or just plug those ratios into a simulator and adjust until a transient response reproduces the observed frequency and rate of decay. I know which I'd do!

If you want to nail the impedance as well, you can make a 2nd measurement adding (say) 1kg extra mass to the C, repeat the parameter extraction, and calibrate what C means in terms of mass (in the case it's a mass spring system, you don't tell us), and therefore the other parameters. 

A version of this was very popular for audio hobbyists building their own loudspeaker enclosures. To extract the loudspeaker parameters, first measure their natural resonant frequency and the damping, then add the extra mass of a bit of plasticine or blu-tak to middle, and repeat.