In one variation the mechanism is thought to be as follows: 
The dichromate oxidizes the mercury, forming a layer of mercury oxide.  In the process the dichromate is reduced to the chromium(III) ion.  The oxidized layer on the mercury reduces the surface tension of the blob and the blob flattens out coming in contact with the iron nail.  Then the mercury sulfate oxidizes the iron to the iron(II) ion, and in the process is reduced back to metallic mercury. Once there is no oxide coating left on the mercury blob, the surface tension increases and the blob rounds up and loses contact with the nail, ready to start the process over again.
The net reaction is that the dichromate oxidizes the iron.  This favorable reaction drives the mercury oxidations/reductions and the oscillations in shape.  When the dichromate is fully reduced, the reaction stops.
There may be other mechanisms involved, however.  Lin et al. appear to report that the oscillations occur without the presence of the oxidizing agent, though the mercury does not appear to get an oxidizing layer on it and the oscillations are much weaker.
An electrical double layer forms between the surface of the mercury droplet and the electrolyte solution. At rest this layer is uniform. When the iron tip is introduced, a redox reaction starts in which iron is oxidized to the ferric ion and the oxidizing reagent is spent (e.g. when hydrogen peroxide together with hydronium ions is reduced to water). Because the oxidation only takes place in the vicinity of the tip, and the reduction process covers the whole droplet surface, the surface tension is no longer homogeneous; this results in the observed oscillations.
Although this reaction is mediated by changes in surface tension, it is very similar in mechanism to other chemical oscillators such as the Belousov–Zhabotinsky reaction, which has several intermediate redox reactions driven by the oxidation of malate by bromine.