The purpose of the counterpoise is to provide an efficient means of conducting the return RF current of the antenna. In the absence of an efficient counterpoise, the current will return through the lossy earth or poorly quantified paths such as coaxial cable shields.
An efficient counterpoise will carry the same current as the primary radiating element. A quarter wave vertical antenna with an efficient counterpoise has a feedpoint impedance of ~34 ohms. We can use Ohm's law to calculate the current:
where I is the current in amps, P is the power in watts and R is the resistance in ohms.
So a 100 watt transmitter fed into a 34 ohm antenna will result in ~1.4 RMS amps. When you have three counterpoise radials, the maximum current in each radial is 1/3 of the antenna current or ~0.47 RMS amps in this example. And generally this current density is dominant near the base of the antenna and becomes less as you move away from the base.
Before selecting a wire gauge, the construction of the wire must be considered. A pure copper wire is easier to analyze than a copper clad wire. In recent years, doorbell type wire, for example, has largely become copper clad. Because the specifications of the copper cladding is rarely published, you should shy away from cladded wires if you have other options since the steel core of the wire is quite lossy if the copper plating cannot carry the majority of the RF current.
Then finally, the gauge of the wire and the frequency of operation determines its losses. As the frequency rises, the current becomes more crowded on the surface of the wire due to skin effect. As an example, a 24 gauge copper wire that is 2.5 meters long (about 1/4 wavelength at 28 MHz) has a DC resistance of ~0.2 ohms but at 28 MHz it has an AC (RF) resistance of ~ 2.2 ohms. You can calculate the AC resistance for any wire size with various online calculators such as this one.
Since you are attempting to increase the efficiency of your antenna, a good goal is to keep the total AC resistance of your counterpoise to less than 1% of the ideal feed point impedance. From the earlier example, 1% of 34 ohms is 0.34 ohms. Since you are using 3 wires, each wire should be 1 ohm or less (0.34 * 3). So the 24 ga wire for 28 MHz would not meet this goal while an 18 gauge, or larger, wire would.
In general, more than 3 radials are desirable for a ground mounted counterpoise. You can see from the example, that as you install more radials, it is possible to also effectively reduce the gauge of the wire.
It has also been empirically determined that ground mounted radials do not need to be tuned to a 1/4 wavelength to be effective. Generally more shorter radials are better than a few 1/4 wave radials.
With all of this being said, don't suffer from analysis paralysis. Any radials of any gauge wire of any construction is generally better than no radials at all.