It's hard to say when the first ironclad warship was proposed, as many people came up with the idea of putting iron plating on warships at many different times. For example, the "atakebune" warships of the 16th century Japanese warlord Oda Nobunaga allegedly had some form of iron plating on their superstructure. During the early 19th century different people talked about the idea of iron-plated warships, but there was little impetus put into their creation until the Crimean War showed the strength of a type of naval artillery called the Paixhans gun.

Developed by French artillery officer Henri-Joseph Paixhans, the Paixhans gun was the first naval cannon to utilize explosive shells. Shells had been used by land artillery for quite some time, but due to the inherently dangerous nature of shells (they are essentially bombs, after all), they had been limited to use in howitzers and mortars. Howitzers and mortars are designed to lob their shots or shells at a high trajectory, and don't need to fire at as high of a velocity as the sort of low-trajectory cannons that were used in ships at the time. Using shells in high velocity cannons was rather risky. Paixhans was able to develop a new fuse mechanism and cannon that made it safe to use explosive shells on naval ships.

The Paixhans gun was developed in the 1820s and installed on French ships by the 30s, but its first use against enemy ships occurred during the Crimean War. At the naval Battle of Sinop during November 1853, which was the engagement that properly started the war, Russian ships with Paixhans were able to destroy their Ottoman counterparts with ease. Sinop made it clear that wood vessels had no real way to counter these shells, which would lodge themselves in their targets before exploding.

France and Britain declared war on Russia after the Battle of Sinop. The French Emperor Napoleon III, who did not want to risk seeing his ships go the way of those Ottoman ships when attacking Russian coastal fortifications (witch also had Paixhans), ordered the construction of armoured floating batteries, assigning the task to the inspector general of naval construction Garnier, a naval engineer named Guieysse, and Commander Favé, an artillery officer. A floating battery is a vessel with very limited mobility (they might be able to move at a few knots in calm seas and were often towed by other ships) but heavy armament. Floating batteries had been used before, but Napoleon III specified that these were to be designed to be able to deal with enemy shell-fire. He proposed that they use chests full of shot (the solid round projectiles fired out of muskets or cannons), but experiments conducted proved that iron plates backed by wood to be superior to cases of shot when dealing with both shells and shot. The French shared their plans with the British admiralty, and in 1855 the ironclad coastal batteries Lave, Tonnante, and Dévastation helped destroy Russian forts at the Battle of Kinburn with minimal causalities. Britain's ironclad batteries arrived too late to see any action.

During the 1850s the French sought to build up their navy with the construction of many steamships of all classes, but due to improvements in naval guns and the lessons of the Crimean War, the naval commission tasked with this naval construction decided to halt work on wooden ships-of-the-line and instead try to design ironclad seagoing warships (a process that would require lots and lots of men and experimentation). This culminated in the Glorie, designed by the naval architect Henri Dupuy de Lôme and his staff.

Rather than crediting one individual, I'd say that the main driving force behind the ironclad was the French Navy as a whole. Given his heavy involvement in the process, though, I suppose one could say that Emperor Napoleon III deserves a fair bit of the credit.

I can't say that I know much about the history of ironclads in the Royal Navy or other navies besides that of the French. If someone does, feel free to elaborate.

Source:

Baxster, James P. The Introduction of the Ironclad Warship. Naval Institute Press, 1933.