The modern use of the word was spawned by a book by Thích Nhất Hạnh[0] called The Miracle of Mindfulness. It was then codified into clinical practice by (among others) Marsha Linehan, who incorporated the book's ideas (in combination with behavioural therapy) as dialectical behavioural therapy, which was the first evidence-based treatment for borderline personality disorder[1]. Since then it's been a prominent feature of popular psychology (and real psychology, though in a more restricted context).
[0]: https://en.wikipedia.org/wiki/Th%C3%ADch_Nh%E1%BA%A5t_H%E1%B... [1]: Linehan M., Cognitive-Behavioral Treatment of Borderline Personality Disorder