Why is the VOR reference and variable signal modulated at 30 Hz?
In VOR early days, the 40s, the stable phase locked loop frequency generator was not available. A common frequency reference was the very stable public AC power grid frequency. The angular velocity of a synchronous motor is locked on the power source frequency, such motor can be used to actuate a variable resistor or capacitor to generate a stable wave which frequency is locked on the power supply frequency.
A VOR relies on two 30Hz signals: Reference signal and variable signal (more on the VOR principle here). The US grid is 60Hz, a synchronous motor with two pairs of poles turns at 1800 rpm, that is 30 rounds per seconds.
The reference modulated the frequency of a 9,960Hz subcarrier (332x30Hz) with a modulation index of 16, meaning the subcarrier swinged between 9,480Hz and 10,440Hz. Here is the description of the mechanical assembly used:

A tone wheel is driven by an 1800 rpm (30 rps) motor; and on this
wheel are 332 teeth which are arranged in a slightly staggered manner
so as to impart a cyclical variation between 9480 Hz and 10,440 Hz
with the rotation of the motor. The frequency modulated signal from
this tone wheel is then used to amplitude modulate the VOR carrier
signal.

(Source: A comparative analysis of area navigation systems in general aviation, by Stephen Malcolm Dodge, NASA -CR-132504, 1973.)
The motor also drove the rotating dipole broadcasting the variable signal:

Early conventional VOR with rotating beacon, source
Here's what the rotating/modulating assembly looked like for a Collins VOR station:

Motor, tonewheel and dipole of a Collins conventional VOR, 1954. Source: Collins
A single motor used to generate both the reference and the variable signal ensured the two signals remained coherent, and therefore allowed a simple phase angles subtraction to get the bearing.
If the VOR had been designed in Europe, the same motor would have turned at 25 rps, and the signals frequency would have been 25Hz instead of 30Hz. There comes @Sanchises question below, about the consequences of the different grids on the VOR compatibility around the world... While the motor angular velocity is the first to be impacted by a different power supply frequency, harmonic interference must also be considered, for example:

In the ILS there are two reference frequencies at 90 and 150Hz, also chosen as multiples of 30 and remote enough from harmonics of line frequency (60, 120, 180Hz, ...). However 50Hz harmonics include 150Hz and interferences may occur on a 50Hz grid.

Modern solutions to solve both the angular velocity problem and the harmonic interferences problem are straightforward, but they were not 60 years ago. It seems designers at this time chose to recreate a 60Hz source locally and dismissed 50Hz AC for the related portion of the equipment. I got this schematics from a 1968 issue of the French aeronautical navigation engineering group bulletin:

Mechanical generation of 90Hz and 150Hz signals in an early ILS. Source: STNA bulletin, page 88
A 1,800 rpm synchronous motor is used to generate mechanically the two frequencies. But France has opted for a 50Hz grid. A static inverter is used as a local 60Hz power supply. As mentioned in the source document, the inverter frequency is stabilized with a 1% accuracy.