The answer is not very complicated: the mark of the beast is his "New Covenant."
In other words, the false Christ will "write" his law on the minds and hearts of his followers in the same way that Jesus Christ (in the bona-fide New Covenant) had written his law on the minds and hearts of his followers (please see Heb 8:8-12 and Heb 10:16-17, and then compare to Deut 6:5-8 where the right hand is equivalent to the heart in the context of writing laws on hearts and minds).
Unlike the New Covenant of Jesus Christ, however, the Antichrist's "New Covenant" will be both political and religious. Thus for those who possess the name of the beast (Rev 13:17), they embrace the Antichrist as their religious leader (please see Rev 14:1 in this regard), while those with the number of his name (Rev 13:17) embrace him as their political leader. In either case, both categories fall under his economic power (since one can neither buy nor sell without the mark), which unites the religious and political categories. Thus possession of the mark indicates that you are "sealed" in the kingdom of the beast; that is, you are part of his New Covenant of blessing.
Now what does the 666 signify? If the Anitchrist is publishing his law on the hearts and minds of his followers, then this New Covenant is comprised of 613 Mizvot (Law of Moses for Jews) in addition to fifty-three "Parashat Mishpatim," which are 23 imperative commandments and 30 prohibitions found between Ex 21:1 and Ex 24:18, and which are binding on non-Jews. (Please click here for more information.) In other words, this New Covenant (613 + 53 = 666) is for both Jews and Gentiles (written on minds and hearts!), and is in fact which unites them under one leader. It is a play off of the real New Covenant, which unites Jews and Gentiles -- please see Eph 2:14-16, where the "two groups" in context are Jews and Gentiles in the real New Covenant. In other words, the 666 represents the perfect (number of man x 3) unification of man independent of God, which is why the 666 is actually the "law" of the Antichrist (or better his "New Covenant"). This antichrist therefore will be seen as the mediator and unifier of Jews and non-Jews (which would include Muslims), where no one in past history has been able to achieve in the political sense. In other words, Muslims will also embrace him as a prophet from God.
Of course the false Christ has his false prophet, whom I refer as the "Anti-Elijah." Please click here to compare and contrast the meaning and chronology of this herald vis-à-vis the Biblical concept of the New Covenant. The false prophet will anoint the head of the beast (at the beginning of the seven year tribulation) and thereby heal his fatal head wound, and thus the Antichrist will be "resurrected by God." Again, we are using copy-cat imagery from the bona-fide New Covenant of Jesus Christ to show how the Antichrist is the Satanic counterfeit...
In summary, the appearance of the false Christ (Antichrist) with the false prophet (Anti-Elijah) will herald the golden age of the history of the world, which of course is the Satanic version of the New Covenant. That is, the very first half of the Tribulation (three-and-one-half years) will be the greatest prosperity the world has ever known. Please click here for more information.