In my name, because ye belong to Christ. The reading adopted in the Revised Version is, ἐν ὀνόματι ὅτι χριστοῦ ἐστέ: literally, in name, that ye are Christ's; or, because ye are Christ's. The force of this observation seems to be this: "If he who gives you a cup of water to drink in my Name, and out of regard for me, does well, and shall be rewarded of God, much more shall he be rewarded who casts out devils in my Name."
The disciples are thus taught that it is contrary to the whole spirit of Christianity to disparage works of beneficence, or to suggest unworthy motives for them (see 'Speaker's Commentary,' in loc.).