we go every year last weekend of February. If you have one week all year I think that is the best. For January if it is happens to be a deplorable year, snow coverage still might be light, (although that would be doubtful for LCC, could be possible elsewhere), and stuff closed yet. Also, January is cold and dark. In March the sun starts coming on strong and high and could ruin conditions fast. Even in the end of February, you can feel the sun starting to destroy good snow on south-facing slopes. End up getting big moguls and powder gone bad quick on a sunny day. It can and does get warm even the end of February. Also, I think March might tend to be more crowded with spring breakers and fair weather skiers. Skip Washington's birthday weekend.
But this year will be my 3rd in a row where I will be getting out West for an extra trip on 2nd weekend of December, Aspen, and an early season trip is for sure a great way to start the winter.
Last 2 years I skied Alta/Snowbird 2nd weekend of December and even though the snow was off to much below average starts, i was saved by the bell, and skiing was just fine! Of course, the time to go, is when you can!