As others have mentioned the Ottoman Empire did sort of have its own Meiji Restoration: the Tanzimat, which did generate tremendous progress during the mid-19th century. A positive POD for the Ottoman Empire would be somehow avoiding the reign of Abdulhamid the 2nd. He was a conservative ruler whose reign is associated with the end of the Tanzimat and the first constitutional era in the 1870s. Reform would not be revived until the Young Turk revolution in 1908. Thirty crucial years were thrown away thanks to him. If a more pro-reform Sultan was in power instead of him the Ottoman Empire could have continued to make progress to the end of the 19th century and beyond.
Another significant problem in the era was the conflict with Russia, the Russo-Turkish war of the 1870s was a disaster, but as has been explored in another AH thread on these boards, that might have gone the other way. So a good way to have the Ottoman restoration succeed might be to have a different Sultan under whom the Tanzimat never ends and different tactics are employed allowing a victory or at least less severe defeat in the war with Russia. Strengthening the Tanzimat's success might also have the effect of reducing the OE's problem with ethnic nationalism, as one of the major goals of the reform era was Ottomanism: the creation of a Ottoman civic nationalism which would allow all the subjects of the empire to share in one national identity and remove the Millet system.
Still, it needs to be said that a successful Ottoman reform effort is a lot harder than a Japanese one. The OE was surrounded by hostile stronger powers and not isolated like Japan. Foreigners had a lot more influence in the OE, they came to control the empire's finances and, as another poster mentioned, engineered the removal of reformist viziers. The OE has a much bigger problem with internal rebellions, which were often supported by foreign powers. Significantly the Japanese were far less likely to oppose the commands of the emperor whom they considered divine and felt they must always obey, whereas the Ottoman sultans faced much more internal opposition and were repeatedly removed when they attempted to make major changes.