Pali
In 1881, Thomas William Rhys Davids first translated sati into English mindfulness in sammā-sati "Right Mindfulness; the active, watchful mind". Noting that Daniel John Gogerly (1845) initially rendered sammā-sati as "Correct meditation", Davids explained,
sati is literally 'memory' but is used with reference to the constantly repeated phrase 'mindful and thoughtful' (sato sampajâno); and means that activity of mind and constant presence of mind which is one of the duties most frequently inculcated on the good Buddhist."
Henry Alabaster, in The Wheel of the Law: Buddhism Illustrated From Siamese Sources by the Modern Buddhist, A Life of Buddha, and an Account of the Phrabat (1871), had earlier defined "Satipatthan/Smrityupasthana" as "The act of keeping one's self mindful."
The English term mindfulness already existed before it came to be used in a (western) Buddhist context. It was first recorded as mindfulness in 1530 (John Palsgrave translates French pensee), as mindfulnesse in 1561, and mindfulness in 1817. Morphologically earlier terms include mindful (first recorded in 1340), mindfully (1382), and the obsolete mindiness (ca. 1200).
John D. Dunne, an associate professor at University of Madison-Wisconsin whose current research focuses especially on the concept of "mindfulness" in both theoretical and practical contexts, asserts that the translation of sati and smṛti as mindfulness is confusing and that a number of Buddhist scholars have started trying to establish "retention" as the preferred alternative.
Bhikkhu Bodhi also points to the meaning of "sati" as "memory":
The word derives from a verb, sarati, meaning “to remember,” and occasionally in Pali sati is still explained in a way that connects it with the idea of memory. But when it is used in relation to meditation practice, we have no word in English that precisely captures what it refers to. An early translator cleverly drew upon the word mindfulness, which is not even in my dictionary. This has served its role admirably, but it does not preserve the connection with memory, sometimes needed to make sense of a passage.
In What Does Mindfulness Really Mean? A Canonical Perspective (2011), Bhikkhu Bodhi points out that sati is not only "memory", but also a "lucid awareness of present happenings". Also, he quoted the below-mentioned comment by Thomas William Rhys Davids as "remarkable acumen":
But as happened at the rise of Buddhism to so many other expressions in common use, a new connotation was then attached to the word, a connotation that gave a new meaning to it, and renders ‘memory’ a most inadequate and misleading translation.