If you have the luxury of coming on a trip at any time of year (rather than only when your boss will give you time off), there are several things to consider.
Temperature
If you like it warm, come at the peak of summer in July/August, when the land is green and lush. For crisp, sunny days, Sept/Oct is ideal, once the heat of summer has passed. An early season trip in June will allow you to see Mongolia's incredible diversity of wildflowers. If you really don't want to see anyone else, a winter trip is for you.
Bugs
Mongolia's reputation for having lots of annoying insects is rather over-rated in our opinion. But if you really don't like the odd mosquito having a nibble, come in Sept/Oct when temperatures start to drop.
Festivals
Mongolia has several festivals that you should try to see if possible, ranging from the small and quirky Yak Festival, through to the full spectacle of Naadam in Ulaanbaatar. Several of our trips are timed so they include one of the festivals (eg the Eagle Festival), or so you can include one as an add-on before or after your main trip.
Our treks generally run between June and early October, but we can arrange trips at any time of year.