Symbolism, Imagery, Allegory
Midwinter's Day isn't just Will's birthday; it's also the winter solstice. This means it's the darkest day of the year, the day when we see the sun the absolute least before the days start growing longer again. It is, if you will, the day darkness most encroaches upon the light. In a book about an ongoing battle between the Light and the Dark, this is pretty significant.
This is also the day that Will starts stepping into his powers. Because this happens on Midwinter's Day, we get a little clue that these powers aren't going to be just any old superpowers—they're going to be connected to pushing back the dark… or the Dark, as the case really is.
For more on the significance of light and dark (and Light and Dark), though, read up on the topic elsewhere in this section.