What was the name of the great compassion meditation the Buddha did for a significant part of his daily routine? Didn't the Buddha do some kind of great compassion meditation  for a significant part of his daily routine while teaching? If so, what's the name of it and what did it involve?
Compassionate meditation aka (Karuna meditation) is a form of meditation to develop compassion for others and ourselves. Buddhism holds a significant amount of importance towards compassion. Practicing Loving Kindness, using Mantras, Mudras, Affirmations are ideal for this form of meditation.
Steps to be followed while considering this form of meditation are:

Find somewhere quiet where you will not be disturbed. Sit with a good and comfortable posture. Place your feet shoulder width apart. Sit with a straight but relaxed spine.
Close your eyes and spend sometime with mindful breathing. This helps in promoting relaxation and focus.
Bring to mind a person who has currently or is suffering. Begin with people for whom  you feel the most sympathy. Sincerity is everything in Karuna meditation. Therefore, start with someone you love.
Consider the struggles this person is facing.
Wish the individual freedom from suffering. Wish for them to be happier, healthier, more fortunate and more successful. Or if a person is ill you wish for them to get healthier!
If you feel any conflicting emotions as such, just observe them don't be judgemental about the situation.
Observe the feeling of compassion. Be mindful of it. How does the feeling feel in the body and mind?. Are there any obstacles in the way to genuine compassion? Compassion is a feeling. And feelings are a source of energy! Connect with the inner energy of compassion. And thats the gateway towards feeling like a Boddhisatva.
Repeat the script meditating on different people.
Finish with a wish to free all sentient beings from suffering.
Sometimes, we may not feel compassion for others and sometimes even for ourselves, if that is the case, we need to start off with forgiveness.

And thats where Meditation on forgiveness starts.