There is a paper about this in the Bulletin of Volcanology.
Cooling rate seems to be the main parameter controlling column size (faster cooling yielding to thinner columns). According to the authors, cooling rate is itself controlled by two non-independent factors: (1) the geometry of the lava body, and (2) the chemical composition of the lava.
(1) Geometry of the lava body influences the boundary conditions for cooling, depending on the emplacement setting (air, host rock, ice...). In the case of a lava lake (figure 2b), lava close to the lake walls will cool faster than lava at the centre of the lake, resulting in thinner columns. Thickness also plays a major role, as shown by figure 10: thicker lava bodies cool slowly, generating thicker columns.
(2) Chemistry plays a role (figure 11): felsic (SiO2 -rich) lavas tend to have thicker columns than mafic lavas. It seems a bit counter-intuitive, as felsic lavas are usually emplaced at a "low" temperature and should thus cool faster than a basalt emplaced at 1200 °C... But felsic lavas are more viscous and tend to form thicker flows -> back to (1).