These things are zero to low maintenance. If it is an outdoor high voltage type (say 33kV) then its insulator body may need cleaning.
If however you are trying to figure out when to replace it (as these things degrade over time as they shunt voltage spikes to ground) then I am not aware of any way except periodic visual inspection (assuming they arrestor is fitted with some device or mechanism to indicate it has failed and removed itself from the circuit and hence needs to be replaced).
You may find more information in the data sheets and application notes relating to your specific lightning arrestor.
If it is a HV type as you suggest, then the Doble Insulation Power Factor (dissipation factor) test can detect degradation in the arrester. It usually shows itself as high watt losses as compared with typical values (or compared with other identical units in the same installation) in Doble's database.
If the arrester is severely degraded, sometimes a megger test (DC insulation resistance) can detect it, but often by that point the arrester fails catastrophically.
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To get the right answers, first you need to ask the right questions.
It depends on the type of surge arrester; older, gapped silicon-carbide or more modern gapless metal oxide (MOV). The most meaningful tests would be grading current and power (or dissipation) factors. As another poster mentioned, Doble does have a lot of information (most of which they gain from manufacturers through their Doble Clients).
The important part is to make comparative measurements under identical test conditions; ideally, initial benchmark measurements at installation. Subsequent periodic measurements are then made for comparison and significant changes are cause for further investigation.
If one were to monitor sound MOV arresters, at normal system operating voltage, they are almost entirely capacitive. As voltage is increased up to the conduction starting voltage, they become more resistive. In full conduction, the arrester is entirely resistive, with current and voltage in phase.
Remember, surge arresters, regardless of type, are not designed to protect against power frequency overvoltages. They are designed and intended to protect against impulse overvoltages. Their power frequency overvoltage capability is defined by TOV (temporary overvoltage) rating for gapless MOV and reseal voltage for gapped silicon-carbide.