Proposed choice of date:
Short answer: November 5th
Long Answer: I'm going to stick to nonleap years for a moment. So exactly one day of the week repeats 53 times. In 2009 this is a Thursday, so I'm going to say Thursday from there on out. If it is a different nonleap year, just replace Thursday by whatever day of the week happens on Jan 1st that year. You will get the same answer for the date.
Since every month has 4 or 5 of Thursdays, there exactly 7 months which have 4 Thursdays. So at least one quarter of the year has one or less 4-Thursday month. It turns out this is uniquely the 4th quarter, and the month is November. Further, it can't be November 1st or 2nd, because their days of the week happen 5 times. So there must be exactly one Thursday with a single digit date. This turns out to be November 5th.
November 5th is also the first time Thursday occurs on the same day of the month for the third time (February 5th and March 5th were also Thursdays.) And it's the farther Thursday in November from Thanksgiving.