Approximation of Hilbert Transform Using Very Short Hilbert Transform FIR / IIR Filter
This is more like an extended comment to chart the possible answers.

Hilbert transform is a frequency domain 90-degree phase shift of the signal. It has an antisymmetrical impulse response around time = 0. You specify that the approximation shall be causal (EDIT: this requirement has since been removed), so I think you need to reference the phase shift to that provided by another filter. If we denote the two filters F1 and F2, some usable alternatives are:

Case | Filter F1    | Filter F2           | Phase diff     | Magnitude frequency responses
-----+--------------+---------------------+----------------+------------------------------
  C1 | pure delay   | antisymmetrical FIR | exactly 90 deg | F2 has ripple
  C2 | pure delay   | all-pass IIR        | approx 90 deg  | No ripple
  C3 | all-pass IIR | all-pass IIR        | approx 90 deg  | No ripple


In all those cases you have two filters whose phase frequency responses have a phase difference of exactly or close to 90 degrees over the desired frequency band.

If the filter outputs are used "in quadrature" (summed, with the other multiplied by the imaginary unit) to create an analytic signal, any phase frequency response ripple will become magnitude frequency response ripple of the composite negative frequency removal filter.