Though I don't have inside information, it's clear that the notes in Asterisque 100 (1982) by BBD gave the first formal definition of t-category and t-structure on a triangulated category (section 1.3). Since the nearest t-word involved is "triangulated" and no direct rationale is offered for the language introduced, that probably suggested the terminology here. Of course, someone else may have a more subtle theory.
P.S. Reviewing the original text further, I see in their Example 1.3.2(1) the
suggestive word tronque, which converts me to Donu's viewpoint. It would have simplified things for the reader of BBD just to start with a definition of categorie tronque (abbreviated t-category).