Walpola Rahula (1907–1997) was a Sri Lankan Buddhist bhikkhu, scholar and writer. When he became the Professor of History and Religions at Northwestern University in 1964, he also became the first bhikkhu to hold a professorship in the West. His introduction to Buddhism, What the Buddha Taught, is considered a modern Buddhist classic. In the book, Rahula writes,
The word meditation is a very poor substitute for the original term bhavana, which means ‘culture’ or ‘development’, i.e., mental culture or mental development.” 
I think Rhaula’s statement would also apply to the word “mindfulness.”
I don’t know what Rahula means by “original term,” but I do recall either reading or hearing that bhavana was the term most often used by the historical Buddha in reference to meditative discipline. According to Alan Sponberg, the term is “certainly the broadest in its semantic range.” 
Rahula called bhavana “mental culture.” Amadeo Sole-Leris wrote that bhavana is “to cultivate and develop the vast potential of the mind in order to overcome the unsatisfactory nature of the internal and external circumstances in which we find ourselves.”  Someone else (I don’t remember who) called it “creative control of the mind.”
I like the word development because I feel it accurately describes the process. In Buddhist practice we develop our innate potential for well-being and happiness. We can also say that bhavana in all its various forms is a system for training the mind. Buddhism teaches that an undisciplined mind is disturbed by circumstances such as gain or by loss, comfort or hardship, and is attached to transient things, all of which invites suffering. We want to train the mind so that we can learn better how to use reason and wisdom to deal with life’s challenges, and so we can break the habit of seizing and clinging, and this brings freedom from suffering, insofar as we can accept sufferings as they come without losing inner peace.
The wild, untrained mind that we often dub “monkey mind,” can be pacified, the restless monkey brought under control. However, if we limit the broad range of practice bhavana covers to only the mind, then it is a poor substitute for what the Buddha taught.
What the Buddha sought to achieve personally was nothing less than total transformation of his entire being. That meant body as well as mind. I wonder if sometimes we don’t tend to focus on the mental health aspect of meditation and mindfulness and neglect the physical health side. Gautama Shakyamuni was called the Great Physician and his teachings the King of Medicines, not only for his psychology of mind. After the Buddha’s passing, a great tradition of healer-monks emerged, and this tradition is still upheld today in the system of Tibetan Medicine and healing.
Here is a wonderful explanation on the relationship between mind, body, and bhavana from Tulku Trondup, a prominent teaching in the area of Buddhist healing, that I found in his book Boundless Healing:
Mind and body are intimately connected, and the relationship of mind to body in meditation is very interesting. When we see the body as peaceful and beautiful, who or what is creating these feelings? The mind is. By creating peaceful feelings in the body, the mind is absorbed in those feelings. So although the body is the object to be healed, it also becomes the means of healing the mind – which is the ultimate goal of meditation.” 
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 Walpola Rahula, What the Buddha Taught, Grove Press, 1974, 68
 Alan Sponberg, “Meditation in Fa-hsiang Buddhism”, Traditions of Meditation in Chinese Buddhism, Ed. Peter N. Gregory, University of Hawaii Press, 1986, 19
 Amadeo Sole-Leris, Tranquility and Insight An Introduction to the Oldest Form of Buddhist Meditation, Shambhala, 1986, 21
 Tulku Thondup, Boundless Healing Meditation Exercises to Enlighten the Mind and Heal the Body, Shambhala, 2000, 12