befriend

To befriend somebody is to become a friend to them, especially somebody who needs one. A deliberate act rather than something that just happened.

That deliberateness is the whole word, since ordinary friendships are not described this way. It belongs to charity and to stories, so befriending an elderly neighbour or befriending a stray dog are both natural. Volunteering schemes use it as a formal term, and a befriending service is a real thing. The word is slightly formal and warm, and it very often carries a hint that the friendship was one-sided kindness.