In the meditation tradition, mindfulness is a specific quality of consciousness.
"...a technique in which your attention is not distracted from whatever it had as its original object of meditation..." —Tsong Khapa, The Great Treatise on the Stages of the Path
"Mindfulness... prevents your attention from wandering from the object of meditation..." —Vasubhandu
"...the foundation of cultivating concentration is mindfulness which does not forget the object." —Tsong Khapa, The Great Treatise on the Stages of the Path
"[Mindfulness] refers to how your attention is fixed on the object of meditation and brings it to mind clearly without even the slightest distraction. If you are distracted, you lose your mindfulness to the extent that you are distracted." —Tsong Khapa, The Great Treatise on the Stages of the Path
"The erring elephant of your mind is securely bound by the rope of mindfulness to the sturdy pillar of the object of meditation and is gradually controlled with the iron hook of intelligence [vigilance]." —Bhavaviveka, Heart of the Middle Way