Usable Mercury cyclers exist.
Given the large inclination of Mercury, we want the encounter to happen along the line of apsides, thereby requiring a quasi-periodic stationary cycler. The simplest type of cycler in this family is a high periapse fly by ellipse, ideally close to a Hohmann transfer orbit. This requires that the planets' synodic period is a simple fraction of the orbital period of either of the planets. Furthermore, the orbital period of the cycler must be close to the previously stated quasi-period, divided by an integer.
Interesting quasi-periods include:
(Synodic periods, inaccuracy in degrees)
As for actual appearance of the planetary alignments, a usable launch window would in worst case appear within a decade in the Mercury-Earth system.
The 22 and the 41 synodic period cycler are the two most interesting. The 22 (7 year) orbit offers a trajectory suitable for reuse 4-5 times, at a relatively low delta-v cost. This is because the orbital period of 0.5385 years, compared to the 0.5327 years for a perfect Hohmann transfer to the ascending node, has a close to minimal vinf.
That is also the case for the even better 41 (13 year) orbit, where the same comparison is 0.61905 years to 0.61853 years for a descending node transfer. (The deviation from a Hohmann transfer is negligible.) A slight boost to a 0.63636 empty 7 year cycle every 46 years reduces the angular drift to 1.635 degrees per century, allowing stability for maybe as long as a millennium.
This type of cycler does not offer two-way transportation, so two cyclers are necessary. (One inbound, one outbound.)