The term "mindfulness" is from a British-Indian translation written in 1921 of the Pali word, "sati" which connotes awareness and attention. The term "mindfulness" however doesn't adequately describe the action and process that's truly meant to be taking place.
"Mind-full-ness" makes the listener imagine they have to use their mind or their thoughts to achieve some goal, when in essence they should be experiencing observation of thoughts and emotions using their awareness. Awareness is our innate human ability to observe our thoughts and feelings, and can be thought of as the silence behind all our thoughts and feelings. Awareness is presence.
Single words that can be used in place of mindfulness to more accurately describe the intended action and process are: PRESENCE, ALERTNESS, SHARPNESS, WATCHFULNESS, ATTENTION, AWARENESS, OBSERVATION, & FOCUS.
The goals of these actions are to cultivate insight and actual transformation moment by moment of thoughts and emotions, in our own inner nature, or "inner space" as we say in HMHB. This is done through awareness training, and by self-awareness in every moment of our lives. Before we talk about the transformational aspect of these actions, where awareness is put to its fullest potential (which is the "beyond mindfulness" aspect we are talking about), let's talk about the insights you can gain simply through awareness training.
If we are truly practicing awareness training actions, then we come to realize through our personal experience, or direct insight that:
1) Thoughts and feelings must be observed every moment if we are to have any control over our inner life (and consequently our external life.)
2) When we get stuck in thoughts and feelings (and believe that we are them, and them alone) is when we are in pain and suffering (and consequently create our own pain and suffering.)