Please read the warnings posted about hiking Mt Washington! It truly is the " Most dangerous weather in the world" ( Google is great for this)
Not to discourage you, but be prepared for the worst. Many people, including experienced hikers, have died on that little hill because they did not have proper respect for how fast the weather can change there. Europeans and western US climbers are not impressed because of its low elevation--but that is a trap waiting to kill. Even though its peak is only 6288, that is more than 1000' above the tree line. People have died within a few feet of reaching a shelter, but couldn't see that far due to the poor visibility.
I was up there last month and the summit conditions were mid 80 degrees and winds at 3mph. On past trips with similar conditions at the base, conditions at the summit have been temps in the 30s and windspeeds in the 60 to 80 mph range.
It is an awesome place--if extremes of weather appeal to you. Just take all recommended safety tips.