Phase-shift filter design
This would be an allpass filter. Except for the trivial case of unity and integer-sample delays, these can't be done as FIR filters and in general an IIR filter is required. However, they are easy to make. The zeroes of an allpass are simply the inverse of the poles (and vice versa). If you have the poles in polynomial form, you can simply flip them to get the zero polynomials. For example a second order allpass looks like this 
$$H(z)=\frac{a_{2}\cdot z^{0} + a_{1}\cdot z^{-1} + a_{0}\cdot z^{-2}}{a_{0}\cdot z^{0} + a_{1}\cdot z^{-1} + a_{2}\cdot z^{-2}}$$
Strict allpass filter have $\left \|H(e^{j\cdot \omega })\right \|=1 $ for all frequencies. You can certainly design approximation using FIR filters if you only need this property for a limited frequency range and if the magnitude doesn't have to be exactly unity.