When the sort of progressive Alexander II was replaced by Alexander III's staunchly conservative regime, the Russian government started viewing Armenians as a subversive and revolutionary element. In addition to that, Russia was less pro-Armenian because it was more pro-Ottoman; there was a general understanding with Abdulhamid - he lets Russian trade pass by uninterrupted, but keeps the Straits closed to foreign (basically: British) warships which would like to threaten Russia through the Black Sea. Between the warmer relations with the Abdulhamid and Alexander III's suspicion of Armenians, Russia had no interest in splitting up the Ottoman Empire. Especially since they wanted to focus on Asia.
Meanwhile, Britain was secure enough in its position that it no longer felt obligated to support the Ottomans; so it indulged its humanitarian streak. Tons of proposals were put forth - to depose Abdulhamid, to intimidate him into radical reforms with a military action, and last but not the least: to simply partition the Ottoman Empire. But all these plans required the cooperation of at least some of the other Great Powers, and they - Russia and Austria and Germany and even France - all rejected it for their own reasons.
As for why Russia gradually became a friend of the Armenians again - that was also a complex issue. First, the reactionary Alexander III was replaced by the more moderate Nicholas II. A new imperial governor in the Caucasus, Vorontsov-Dashkov, managed to win over many of the local Armenians with new, tolerant policies; so St. Petersburg itself became more confident of its Armenian subjects and thus more inclined to champion the cause of Armenians across the border. The semi-reliable Abdulhamid was replaced by a series of Young Turk governments, which continued much of Abdulhamid's friendship with Germany but little of his friendship with Russia...and so on.
Interestingly, Russia never completely settled on the Armenians even as they reconciled; allying with Kurdish nationalism instead was an option and a parallel course all the way up to WWI.