Weather (frozen winters), festivals (Naadam in July) and crowds (Tourists arriving for the Naadam) are the three factors to be considered.
If the Naadam festival is your goal, so summer is the time.
If you are excited by frozen lakes, so winter months are the best.
If comfortable weather, and less crowds are important to you, than autumn and spring are great.
January February March
Weather - Below freezing, lakes and rivers are frozen, mountains are covered with snow.
Festivals - Ice festival on the frozen Hovsgol lake, Tsagan Sar festival, Thousand camel festival.
Crowds - winter tourism is yet to be developed, no crowds.
April May June
Weather - coming out of winter, warming up.
Festivals - preparing and training for the Nadam festival.
Crowds - early birds, not crowded yet.
July August September
Weather - Warm temperatures, hot in the Gobi desert, summer is also the rainy season.
Festivals - July is when the Naadam festival occurs throughout the country.
Crowds - This is when most tourists come to Mongolia.
October November December
Weather - Getting colder
Festivals - Altai Eagle festival
Crowds - Most of the crowds are gone.