It appears the allies of Britain during the war (Ottoman Turks and France) had much stronger reasons for joining the war i.e the Turks were responding to Russian demands to exercise protection over the orthodox subjects of the Ottoman Sultan and the dispute between Russia and France over the privileges of the Russian Orthodox and Roman Catholic Churches in the holy places in Palestine. The only explanation I have come across is that it was to stop Austria becoming involved. Were Austria a military power in the 1850's? Why would Britain join a war just to stop them joining one or was there a different reason all together?
tl;dr Britain didn't get much out of the war?
ninja edit: After some more reading up it appears Russia only accepted preliminary peace terms after Austria threatened to join the war on the allies side. Were Austria really that powerful? Also it seems a lot of the war was to do with countries wanting power of the Danubian principalities, firstly Russia, then Ottoman Turks and then Austria. What is so significant about that area (Modern day Romania)
Around this time, Russia had been looking hungrily at Central Asia. It was preparing to subjugate many of the independent khanates in the area, including Afghanistan, which bordered British India. The British were, needless to say, uncomfortable with the thought of Russian troops marching into the "jewel of the British Empire." In what has been termed "The Great Game," Britain and Russia competed for influence to maintain and break, respectively, Britain's control of India.
To that end, Britain saw Russia as a threat. Allowing Russia to strengthen its position in Europe against the Ottoman Empire would have given Russia significant ability to to maneuver into other areas, including Mesopotamia and Iran, which would also threaten British India. A weaker Ottoman Empire would also be less able to bottle up the Russian Black Sea Fleet. This would allow Russia to gain control in the Mediterranean, which posed a threat to British interests in Egypt, especially... you guessed it, trade to India via Alexandria and the Red Sea.
TLDR: Strengthening the Ottomans against Russia was a way for Britain to keep the Russians from threatening British India.