I very much like this comment by Jon Kabat-Zinn:
Mindfulness is often spoken of as the heart of Buddhist meditation. It’s not about Buddhism, but about paying attention. That’s what all meditation is, no matter what tradition or particular technique is used.
In Asian languages, the word for mind and the word for heart are same. So if you’re not hearing mindfulness in some deep way as heartfulness, you’re not really understanding it. Compassion and kindness towards oneself are intrinsically woven into it. You could think of mindfulness as wise and affectionate attention.