Highly damped IIR/FIR digital filter I want a highly damped highpass filter (damping of at least $2$), with a cutoff somewhere around $1\textrm{ Hz}$ ($51.2\textrm{ Hz}\quad f_s$)


How do i go about designing the filter with adequate control over the damping?


My best guess was to use a standard second order response and frequency transform it, giving 

$$
\frac{s^2 \omega_0}{\omega _0^2 s^2 + 2\zeta\omega_0 ^2 + 1}
$$ 
This gives me the required damping (after using MATLAB to transform it to the discrete domain using Tustin's method with tustin), but the frequency response is terrible (the $\rm dB$ point is much, much higher than it should be). 


Is there any way to solve this? 
Can I design a FIR filter with control over the damping? 
Is there a higher order $s$-domain function that still gives me control over the damping?


Notes: filter order and phase delay are not overly important to my design.
Your second-order high pass transfer function is wrong. You should use 

$$H(s)=\frac{s^2}{s^2+2\zeta\omega_0 s+\omega_0^2}$$

But - as mentioned in the comments - a high damping and a sharp cut-off are incompatible requirements.