As the strategy of all players is immediately effected by second hand not following suit, all terms for this distinguish - of necessity - between discarding a side suit and playing a trump. The latter is always an attempt to win the trick while the former abandons ability to win the trick. In over 6 decades of playing a wide variety of trick-taking card games, the notion of being able to proceed in any manner without knowing second hand's choice has never arisen.
Including terms identified in comments above:
Synonyms for not following suit with a trump:
- overruff or overtrump - when an earlier hand to the trick has already ruffed
- be tapped - forced to ruff in a context where control of the hand is at stake and one would have preferred not to trump
- uppercut - trumping to force opponents to over-trump high (due to some other threat in hand), thus establishing a trump trick in partner's hand through either length or power
Synonyms for not following suit with a non-trump:
- discard (the technical term preferred in writing Rules and Laws)
- slough or sluff
- unblock - discard of specifically a blocking high card in a side suit, to improve communication between hands
- throw off
- cast off
- dispensed with - would apply when discarding a card being held for a purpose that no longer applies, such as in squeeze defense
- sluff and ruff - the circumstance where a loser is eliminated (and extra winner created) because both hands of a partnership are out of the suit led; thus one can sluff the loser while the other ruffs with an otherwise non-winning trump card
In enlivening a narrative, one could consider almost any synonym for "to throw away" as a possible alternative phrasing for the act of discarding another side suit
Another answer proposes the term break. This terminology only applies to games such as Hearts, where the general goal is to avoid taking tricks in order to avoid capturing penalty cards. Though non-standard, many play house rules where certain suits, such as Spades and Hearts in Hearts, cannot be led until they have been discarded. In trick-taking games the term of having "broken" a suit instead refers to having initially (at disadvantage) led it, not discarded it.
The reason for the term is that there is an inherent disadvantage to being the first side to lead a suit, whether from a guaranteed end-play situation or the necessity of giving up on possible safety plays against a bad break in one direction.