I saw this video from styropyro
At 16:00 he uses a tesla coil gun to ignite a metal halide lamp that wasn't starting (i think the ignitor connected to the ballast failed)
How does this work?

Arcs start when the electric field exceeds what it takes to break down (ionize) the gas

Electrc field is basically voltage divided by distance. For example if there are 2 uniform flat conductors 2mm apart, with 1kV between them, the field is 500V/mm. When the conductors are not uniform, the field tends to be stronger around sharp edges or where the conductors (the surfaces which face the other pole conductor) have smaller surface area

The external Tesla coil applies very high voltage, this high voltage over the distances between it and the arc tube electrodes creates a region with high electric field, some of which happens to also be inside the arctube. If it have sufficient field strength, it will ionize the gas near the electrode

With initial ionization present, the discharge will quickly take over the path between the 2 main electrodes

How does this work?

More so, some lamps can be ignited by vigorously wiping the outer bulb with a synthetic cloth! If you do this in the dark, you'll see buffer gas in the burner flashing due to the strong electric field induced, and even small blue corona at metal parts in the outer bulb (nitrogen filled, I assume).

what about the piezoelectric spark form a lighter? Is that strong enough?

It depends. Piezo generator from a lighter certainly generates a few kilovolts, so if you run wires from it to lamp electrodes, likely the burner will strike momentarily. Just clinking the intact lighter nearby the lamp will be helpless, not enough field strength I guess...

At one point I wired up a HPS ballast with a dead ignitor to a piezo ignitor used for BBQs and the HV pulse from it was enough to start the lamp.

The length of arc in air which this PZT ignitor can produce can give you an estimate of the order of magnitude of its output voltage

In uniform field, air requires 3kV/mm to break down with high probability, with some behavior properties :

 - ionization is always a statistical phenomena, dependent also on factors like photoelectric effect and radiation

 - The presence of a nearby surface of a different material, even if it is in itself dielectric, may sometimes facilitate the arc striking in a path from one electrode towards the surface, in a straight line on the surface layer, and to the other electrode

 - Some surfaces of solid materials may have additional insulation breakdown mechanisms such as tracking, which will happen at way lower field strengths. However, the initial starting of tracking is a relatively slow phenomena and unlikely to happen with the fast pulse of a PZT. Air by itself does not have such mechanisms, only the basic breakdown as result of applied field

 - If you are looking instead for a distance which is safe to NOT break down, that would be no more than few 100's V/mm, and that's without any surface related mechanisms. The AC voltage ratings vs. contact opening distance of small low current relays may give you an estimate here

In non uniform field, the electric field will be concentrated near sharp edges. This means that even if your real voltage and distance between electrodes would calculate as lower field strength, there still will be a region of field way above the 3kV/mm. Once a small region is ionized, it is easy for the arc to spread from there to anywhere else as needed to complete the circuit, even where initially the field is weak

The lamp glass wall have up to 10x the permittivity of air, alumina (HPS arctubes) also 10x, and quartz 4x. (abridged explanation : imagine the 1mm glass wall behaving as a 10mm air gap etc)

The atmosphere in the lamp outer is generally vacuum or nitrogen, which have permittivity same as air

With this said, try to imagine if the arc would reach the lamp electrodes if you would zap it as you do, from the same position, if there would be no lamp or arctube or outer envelope there, but just the PZT and bare lamp electrodes at the respective distances (with the extra air distances that represent the arctube and lamp walls)

If it does, odds are it can strike the arc. If it does not, it may still strike it, because there may be additional conditions which reduce the extra "push" the lamp needs in the 1st place