Gods never hold random objects in their hands. So, I believe that both of these two things that "The spirit of winter" is holding have a meaning related to winter.
The dead leafless branch is an obvious symbol. Trees are leafless in winter.
But there is another possibility, that this is not a dead leafless branch at all, but stag antlers. Considering that stags lose their antlers during the winter, this is another perfect symbol of winter...
But what about the hare?
European hares have a prolonged breeding season which lasts from January to August. Peak reproductive activity occurs in March and April, when all females may be pregnant, the majority with three or more foetuses. The height of the breeding season, is known as "March madness", when the normally nocturnal bucks are forced to be active in the daytime. Female fertility continues through May, June and July, but testosterone production decreases in males and sexual behaviour becomes less overt. Litter sizes decrease as the breeding season draws to a close with no pregnancies occurring after August. The Male hares are infertile during September, October and November.
The mating period starts end of January, beginning of February and ends end of July, beginning of August.
The end of January, beginning of February is the beginning of Spring, which is in the Celtic calendar marked by Imbolc, which is celebrated on the 1st of February, but actually falls on the 4th of February, which is the exact mid point between the winter solstice and the spring equinox.
The end of July and the beginning of August is the beginning of Autumn, which is in the Celtic calendar marked by Lughnasadh, which is celebrated on the 1st of August, but actually falls on the 2nd of August, whcih is the exact mid point between the summer solstice and autumn equinox.
The most intensive part of the hare breeding season is known as "The March Madness". This is where we also find spring equinox, which later became Easter...
No wonder hare was in the past seen as a symbol of fertility...
You can read more about the Celtic calendar in my post "Two crosses".
Interesting right? But here is something even more interesting. Right at the beginning of the hare breeding season, in January, is the best time to see the constellation Lepus, hare.
Considering that we know that Pisces (fishes), Aries (ram), Taurus (bull) and Capricorn (goat) all mark significant events in the reproductive cycle of these animals on the yearly solar cycle, I am convinced that Lepus (hare) is not where it is on the yearly solar cycle by chance...
And this is why the spirit of winter holds a hare in his hand :)