Why is 50 Ω often chosen as the input impedance of antennas, whereas the free space impedance is 377 Ω?
The input impedance of certain devices/circuits (transformers) does not neccessarily need to match their output impedance.

Consider a 50Ω (or whatever impedance) antenna as transformer that transforms 50Ω (wire side) to 377Ω (space side).

The impedance of the antenna is not (only) given by the impedance of free space but (also) by the way it is constructed.

So the antenna does match the impedance of free space (on one side); and ideally also the impedance of the circuit (on the other side).
Since the space side's impedance is always the same (for all kinds of antennas operated in vacuum or air), it doesn't need to be mentioned.
Only the wire side is what you need and can care about.

The reason 50Ω or 75Ω or 300Ω or ... is choosen as antenna impedances is because of practical reasons to construct particular antennas/transmission lines/amplifiers with that impedance.

A possible ansatz for calculating the radiation resistance \$R\$ of an antenna is: 

Find an answer to the question: 
"How much power \$P\$ (average over one period) is radiated if a sinusoidal signal of given voltage (or current) amplitude \$V_0\$ (or \$I_0\$) is applied to the antenna?"

Then you get 
\$R = \frac{V_0^2}{2P}\$ (or \$=\frac{2P}{I_0^2}\$)

You get radiated power \$P\$ by integrating the Poynting vector \$\mathbf{S}\$ (=radiated power per area) over the sphere enclosing the antenna.

The Poynting vector is \$\mathbf{S} = \frac{1}{\mu_0} \mathbf{E} \times \mathbf{B}\$ where \$\mathbf{E}\$ and \$\mathbf{B}\$ are electric/magnetic fields caused by the voltages and currents in your antenna.

You can find an example for such a calculation in the Wikipedia acticle about "Dipole antenna", in  paragraph Short Dipole.