Savai Thundermane

If the target creature is an illegal target by the time Savai Thundermane’s reflexive triggered ability tries to resolve, the ability won’t resolve. You won’t gain 2 life.

You can cycle a card even if it has a triggered ability from cycling that won’t have a legal target. This is because the cycling ability and the triggered ability are separate. This also means that if either ability is countered (with Disallow, for example), the other ability will still resolve.

Triggered abilities from cycling a card and the cycling ability itself aren’t spells. Effects that interact with spells (such as that of Cancel) won’t affect them.