Amphibious command ship
In the Second World War, flagships for amphibious task forces were originally aboard cruisers or battleships, but the shock of their guns tended to disable communications, and the command function moved to dedicated 'amphibious command ships. This role of WWII has largely disappeared, with the disappearance of heavy gun batteries that made warships unsuitable platforms for shock-sensitive electronics, as well as the much smaller size of amphibious operations. Any large amphibious warfare ship, such as a Landing Platform Helicopter, will have abundant staff facilities.
Most command ships in use today have a "L" prefix, as they are most associated with amphibious warfare; the U.S. has three, one for each forward-deployed fleet. They are not, however, cruisers.
During part of the Cold War, two ships were designated as a National Emergency Command Post Afloat (NECPA). The idea of getting the President and National Command Authority safely airborne in the E4B "doomsday plane" or National Airborne Command Post (NEACP) is well known.
The airborne command post would be much harder for an enemy to attack than a ship, but the aircraft has much less endurance than a ship. While the command ships, the USS Northampton, modified from a cruiser, and USS Wright, modified from a carrier, would not fall from the sky if they ran out of lubricating oil, once Soviet ocean surveillance satellites and long-range anti-shipping missiles were available, it is hard to imagine that a ship less protected than is an aircraft carrier, with rings of escort vessels and aircraft, would be survivable. They were decomissioned in 1970. 
- Priest, Karl C., Ghosts of the East Coast: Doomsday Ships