From electronics point of view, the simplest part that can be used for this purpose is a resistor across the relay coil. The inductive voltage spike will then have a maximum voltage of I · R, where I is the operating current of the coil and R is the resistor in parallel.
Because relays as switches can tolerate much higher voltages than semiconductors, the resistance can be large and does not waste much energy. In addition, the large reverse voltage means that the current in the relay coil decays faster than it would with a modern flyback diode, resulting in faster switching of the relay.
The inductive spike voltage is also limited by parasitic capacitance in the circuit and in the relay coil. The result will be high-frequency oscillation in LC circuit until the energy dissipates in coil resistance. In some cases this could be an acceptable way of dealing with the inductive spike, if all components involved can tolerate the maximum peak voltage and there are no EMC issues from the oscillation.
To research what methods were actually used, I studied the US patent 2636672A which details the IBM SSEC electromechanical computer. This is a system that uses both relays, vacuum tubes and specialized electromechanical counters in its implementation.
The circuit diagram does include several series RC snubbers. The snubbers are placed across some relay switch contacts, but not all of them and not consistently.
In modern circuit diagrams it is typical to see the flyback protection drawn in parallel with the relay coil that generates the spike. Electrically it is equivalent to have a snubber across the switch contacts that it is protecting.
Later in section 11, the text of the patent refers to arc suppression:
The Figure 36 referred in the text:
If I interpret the description correctly, the relays AM turns on delay relay AMD, which turns off later. This shunts the inductive spike to the parallel RC circuits.
The relays used in the machine are of double-coil type. In the AMD relay the second coil is short-circuited, which will result in slow turn-off due to the low resistance maintaining the flyback current for a longer time. This simultaneously dissipates the energy into coil resistance. In some diagrams, the second coil is connected to a resistor, which would also delay turn-off but by adjustable amount, and simultaneously work as arc suppression.
Examples of double-coil relay connections are seen in Figure 32b, where the coils are driven by vacuum tubes:
In conclusion, at least RC snubbers, auxiliary delay relays and double-coil relays were used for arc suppression in IBM SSEC computer. I found no evidence of any type of diode being used for the purpose in this machine.