There is a historical reason. But it was not a fluke of history, the underlying reason is that energy comes up in non-mechanical (thermal, electric) contexts whereas momentum does not. Derived alternative, newton-meter in SI, did not arise naturally in such contexts, and alternative units, like calories, were used prior to the discovery of the general energy conservation law. It made sense to replace them with named standardized units. Klein's Science of Measurement is a nice historical survey.
Standardization and naming of units happened in two waves. First started during the French revolution at the behest of Taleyrand (known as bishop turned politician who later managed to serve as foreign minister under both Napoleon and Bourbons after Waterloo). French Academy appointed a committee on weights and measures, including Condorcet, Lagrange, Laplace, Monge, and unofficially Lavoisier before he was guillotined, came up with the foundation of SI. At that point only the base units of length, time, weight and temperature received single names. The derived units of area, volume, speed, acceleration, work/energy, torque, momentum, etc., were combinations of the base ones even if they had commonly named units like liters or knots. Future kilogram, called kilograve, was the unit of weight, i.e. force, not mass. Mechanical energy and momentum received equal treatment.
The second wave, started in 1860s and formalized by 1880s in both SI and its competitor CGS, was meant to catch up with developments in thermodynamics and electromagnetism, and gave us ohms, volts, farads, watts, etc. Kilograve renamed into kilogram became the unit of mass, the unit of force was named dyne in CGS (from Greek dynamis - force), and newton in SI. What of energy? Its mechanical manifestation received no new attention, but in 1864 Clausius suggested erg (from Greek ergon - work) to replace calorie as a unit of thermal energy, a remnant of by then discredited caloric/phlogiston theory. It was adopted in CGS. The unit for power, watt, was suggested even before joule, by Siemens in 1882, to replace Watt's own horsepower used to measure output of steam engines. Siemens was an electric engineer. Joule himself was honored by a unit name for determining the mechanical equivalent of heat. Momentum was out of luck.