Kind of like how others have said: avoid major holidays your family celebrates, birthdays, anniversaries, basically anything important that your family considers important--and anytime close if possible. If that isn't possible, try to avoid holidays and birthdays--as those are set.
But really if you're wanting to avoid a date, normally I say to pick a date that avoids the main holidays, at least by a few weeks. I know in my family where a death occurred on Christmas morning (yeah, they got a call expecting "merry christmas" and instead got a "____ died this morning") they kept on remembering that way too long just because of the date. I know it happened, and I wasn't even born when it happened!