What was the name of the great compassion meditation the Buddha did for a significant part of his daily routine?

Great Compassion, Mahākaruṇā, is different than simple karuṇā, or compassion. It is more properly translated as “great responsiveness” and is the activity, or functioning, of Emptiness. Unlike karuṇā—compassion that must be developed—mahākaruṇā is the intrinsic naturing of all appearances, is Buddha-nature, and thus the Buddha naturally manifested this responsiveness in his every word and deed. Technically, this is not the result of any practice, but is, rather, the fruit of enlightenment. See:

The final attribute of emptiness to be mentioned is a quality peculiar to the Buddhist analysis: responsiveness. It is the third and final denominator in the list of categories or aspects by which emptiness can be defined: essence, nature, responsiveness. It appears anomalous, an attribute rather than a category. The third logical category is function, or manifest function, and the attribute found in its stead is responsiveness and its qualifier is all-pervasive. Viewed as a functional attribute of inner space, total presence, and light, the implication is that the dynamic, the intentionality, the purpose of being is compassion, which is a synonym of responsiveness and demonstrable as the responsive aspect of love. It is this compassion that is coextensive with space, the buddha-heart pervading all beings. Viewed as the potential form or manifestation of emptiness, the implication appears to be that every vibration of body, speech, and mind is a form of compassionate energy, nothing excluded. Consider the distinction between responsiveness and compassion. In Dzogchen, compassion is much more than the virtue of loving-kindness. Nor does the word compassion in the Dzogchen context denote its English etymological meaning, “suffering together” or “empathy,” although both these meanings may be inferred. Essentially, compassion indicates an open and receptive mind responding spontaneously to the exigencies of an ever-changing field of vibration to sustain the optimal awareness that serves self-and-others’ ultimate desire for liberation and well-being. The conventional meaning of compassion denotes the latter, active part of this definition, and, due to the accretions of Christian connotation in the West, response is limited to specifically virtuous activity. Responsiveness defines the origin and cause of selfless activity that can encompass all manner of response. On this nondual Dzogchen path, virtue is the effect, not the cause; the ultimate compassionate response is whatever action optimizes presence — loving-kindness is the automatic function of primal awareness. (Excerpt From: “The Flight of the Garuda,” by Keith Dowman, Wisdom Publications, 2003, pgs 37-38)

However, the meditation that Buddha continued throughout his life, the meditation practice that brought him to realization, is explained in the Surangama Sutra, and its results, in the Surangama Samadhi Sutra. In the Surangama Sutra it is explained by Avalokitasvara as being his practice, resulting in the perfection of mahākaruṇā.

Manjushri, in the same Sutra, explains why this practice is different than all others—it has to do with the support used—and asserts that this is the practice used by he himself, Avalokitasvara, the Buddha, and all Buddhas. He explains why this is so, in the case of all Buddhas—because there is no Dharma extant when a Buddha arises in the world, thus, something other than the Dharma is needed to reach full enlightenment since all other meditation supports are discontinuous. I recommend reading the sections in that Sutra: “The Method Of Avalokitasvara Bodhisattva,” and “Manjushri’s Summation”.

There is also a prophecy disseminated by Jamyang Khyentse Chokyi Lodro that prescribes the use of this practice to attenuate the suffering of our current time, and specifically during the years 2026 through 2032.

So, to directly answer your question, the name of the meditation is just Great Responsiveness Meditation, at least as used in the prophecy. A variant of it, called the Yoga of the Supreme Sound of the Four Elements was once used in Dzogchen practices from Tibetan Buddhism, and Bön, where is was lauded as the most efficacious practice leading to the manifestation of great responsiveness, i.e., enlightenment.