When you cheer up a friend who's feeling down, you bolster them. To bolster is to offer support or strengthen.
A bolster is also the name of a long pillow you might use to make your back feel better. And the two uses are not dissimilar. When you bolster your friends, you support them and prop them up, just like the pillow does for your back. When you're trying to bolster your credibility, you find people and/or documents that support you or your view. Bolster efforts to learn this word!