We meet with the same in substance, Matthew 10:42: there the phrase is, in the name of a disciple; here it is expounded, because ye belong to Christ. In my name; upon my account, believing you have a relation to me.
Related Commentaries of Mark 9:41
Mark 9:41
41 For whosoever shall give you a cup of water to drink in my name, because ye belong to Christ, verily I say unto you, he shall not lose his reward.