Although unlikely, the probably one of the best moves Putin could make would to restore the monarchy, as it could allow him to entrench his power and push Russia's territorial expansion. It would also allow him to do this with a veneer of legitimacy.
Firstly, he could claim that the Romanovs are the legitimate rulers of Russia, and had always been, with centuries of Romanov rule, compared with the chaos caused by the Soviet Union. In reality he'd be appointing a pliant figurehead whilst maintaining the power to himself, acting as a sort of Cardinal Richelieu. By reviving the title of "Autocrat of All the Russias" he could claim that his dictates are somehow linked to a higher power. As wild as this may seem to people living in the West, keep in mind that in Russia, Putin is still immensely popular and he could probably get many of his followers to accept this. In addition, Putin could have the Tsar/Tsarina sweep away some of the remaining democratic checks on his power by Imperial decree, in a sense shielding himself from some criticism.
Secondly there seems to have been a revival in interest in Russia's Tsarist past, particularly the more popular rulers such as Peter the Great and Catherine the Great. Palaces have been restored and some rebuilt along with cathedrals and churches. New monuments have been put up to Tsars, including Nicholas II and his family who are considered martyrs by the Russian Orthodox Church. By having a real Imperial Family, he could parade them around in ceremonies tied to the church and appealing to the amsses.
Thirdly, Putin would gain control over the church as the Holy and Governing Synod had its members appointed by the Tsar. Putin would be able to promote church ideology favourable to the government. Perhaps a wise move for further gaining a grip on the masses would be for Putin to then promote religiousity amongt the masses and attacking non-Orthodox churches as "un-Russian".
Fourth and perhaps most important, Putin could claim that the Russian Empire's dismemberment into republics was an illegitimate act perpetrated by the Bolsheviks. That the Tsar is ruler of all the Russias, freeing him up to claim lands in Georgia, Ukraine etc. Also, he could sweep away with pesky "republics" claiming autonomy and replace them with the traditional guberniyas.
Finally, the revival of a nobility could increase the loyalty of the oligarchs and the military to the monarchy (and therefore Putin) by granting them priveleges and titles. Russia has a wealthy oligarchy that seems to love power, money and living lavishly. Many have built homes imitating that palaces of the Tsars and indeed are reminiscent of the boyar families of yore. What more fitting than to have the Tsar grant them titles of nobility and or knighthoods with the prestige of using a title. They'd love to be addressed with their new titles and I imagine many of the oligarchs and their wives would relish the ability to wear opulent costumes bedecked with jewelled orders and tiaras at court balls.
Of course this might seem ridiculous in the West, but the mindset in Russia is very different from that of Western Europe and the Anglosphere. A large segment of society seems to love pomp and military parades along with oppulence and excess. Also, the majority in Russia seem to like strong leaders, and who would be stronger than an Autocratic Tsar. Having a Tsar/Tsarina would free Putin from having to alter the constitution to extend his term-limits. Rather, he'd be serving at the grace of his sovereign, who is appointed by none other than God.
