Scientists (and science fiction writers) have speculated about the possbilities of life under the surfaces or on the surfaces of large moons in our solar system or large exomoons of large exoplanets in other star systems.
So a good place to find any limits on the possible properties of moons that can have magnetic fields is a scientific discussion of the possibilities of life on exomoons.
Like, for example, "Exomoon Habitability Constrained by Illumination and Tidal Heating" by Rene Heller and Roy Barnes, Astrobiology, 2013.
In section 2. Habitablity of exomoons, they discuss the qualities, including mass, necessary for exomoon habitability.
A minimum mass of an exomoon is required to drive a magnetic shield on a billion-year timescale (Ms ≳ 0.1M⊕,
Tachinami et al. 2011); to sustain a substantial, long-lived atmosphere (Ms ≳ 0.12M⊕, Williams et al. 1997; Kaltenegger
2000); and to drive tectonic activity (Ms ≳ 0.23M⊕, Williams et al. 1997), which is necessary to maintain plate tectonics and
to support the carbon-silicate cycle. Weak internal dynamos have been detected in Mercury and Ganymede (Kivelson et al.
1996; Gurnett et al. 1996), suggesting that satellite masses > 0.25M⊕ will be adequate for considerations of exomoon
habitability. This lower limit, however, is not a fixed number. Further sources of energy – such as radiogenic and tidal heating, and the effect of a moon’s composition and structure – can alter our limit in either direction. An upper mass limit is
given by the fact that increasing mass leads to high pressures in the moon’s interior, which will increase the mantle viscosity
and depress heat transfer throughout the mantle as well as in the core. Above a critical mass, the dynamo is strongly
suppressed and becomes too weak to generate a magnetic field or sustain plate tectonics. This maximum mass can be placed
around 2M⊕ (Gaidos et al. 2010; Noack & Breuer 2011; Stamenković et al. 2011). Summing up these conditions, we expect
approximately Earth-mass moons to be habitable, and these objects could be detectable with the newly started Hunt for
Exomoons with Kepler (HEK) project (Kipping et al. 2012).
Their sources for those mass limits are:
Tachinami, C., Senshu, H., Ida, S. 2011, ApJ, 726, 70
Williams, D. M., Kasting, J. F., Wade, R. A. 1997, Nature, 385, 234
Kaltenegger, L. 2000, ESA Special Publication, 462, 199
Gaidos, E., Conrad, C. P., Manga, M., Hernlund, J. 2010, ApJ, 718, 596
Noack, L., Breuer, D. 2011, EPSC-DPS Joint Meeting 2011, http://meetings.copernicus.org/epsc-dps2011, 890
Stamenković, V., Breuer, D., Spohn, T. 2011, Icarus, 216, 572
So Heller and Barnes beleive that the approximate lower mass limit for a habitable planet or moon or other type of world would be about 0.25 times the mass of Earth. Note that they suggest that a world with of mass of only 0.12 Earth mass could retain a substantial atmosphere for a long enough time.
This is a somewhat lower mass than the 0.195 Earth mass suggested by Stephen H. Dole in Habitable Planets for Man, 1964), as the lowest possible mass for a planet to retain a dense atmosphere for geologic eras. I don't know the reasons for that discrepancy.
Returning to the subject of the lowest possible mass world to generate a magnetic field, it may be noted that the smallest worlds in our solar system to have detectable magnetic fields are Mercury and Ganymede.
The planet Mercury has a mass of only 0.06 Earth mass, but has an extremely large iron core relative to its size. It rotates with a period of 58.65 Earth days, much slower than any other planet except for Venus.
The planet Mars has a mass of 0.11 Earth, almost twice that of Mercury, and rotates much faster than Mercury.
Mars lost its magnetosphere 4 billion years ago, possibly because of numerous asteroid strikes, so the solar wind interacts directly with the Martian ionosphere, lowering the atmospheric density by stripping away atoms from the outer layer.
The planet Venus has a mass of 0.72 Earth, about 12 times the mass of Mercury, but a much longer rotation period of 243.02 Earth DAys.
In 1967, Venera 4 found Venus' magnetic field to be much weaker than that of Earth. This magnetic field is induced by an interaction between the ionosphere and the solar wind, rather than by an internal dynamo as in the Earth's core. Venus' small induced magnetosphere provides negligible protection to the atmosphere against cosmic radiation.
Ganymede, the largest moon in the solar system, has a mass of 0.025 Earth, and has a weak magnetic field.
Callisto has a mass of 0.018 Earth, and Titan, largest moon of Saturn, has a mass of 0.0225 Earth. They don't have detectable magnetic fields.
Their lack of magnetic fields may be due to their lesser mass compared to Ganymede. Since all three, and all moons of Jupiter and Saturn which orbit closer than them, are tidally locked to their planets, their rotation periods are equal to their orbital periods.
So the rotation period of Ganymede is 7.154 Earth days, that of Titan is 15.945 Earth days, and that of Callisto is 16.689 Earth days.
So perhaps the slower rotation periods of Titan and Callisto prevent them from having magnetic fields. All of moons of Jupiter closer than Ganymede, and all the moons of saturn closer than Titan, have rotation periods shorter than 7.154 Earth days, but even the most massive, Io, has only 0.6 the mass of Ganymede.
And of course the internal composition of a world, and other factors, will also influence whether it has a magnetic field and how strong it is.
If it is desired that an exomoon have a magnetic field to shield it from cosmic rays and its star's stellar wind, and so to keep it habitable, it is possible that an exomoon might not need its own magnetic field. It could orbit within the magnetic field of its planet and be protected by the magnetic field of the planet.
Such a possibiity is discussed in "Magnetic Shielding of Exomoons Beyond the Circumplanetary Habitable Edge". Rene Heller & Joge Zuluaga.
And anyone interested in the potential habitability of exomoons should check my answer at: