I was born in January; my real name is linked to a nature/plant-name (it is not Holly or Ivy!) but isn't linked to any season.
I really like January as a name. If my parents had named me January, I would have happily lived with the name. I shall disagree with hermion, and optimistically say that January is the month of new beginnings; of looking ahead; of the new, and dreaming of what is to come.
I your DC is born in January, they'll have £££s to spend in the January sales
from any birthday money received.
In terms of NNs, Jan is an obvious one, but you could have Jay, Janey, Jamie/Jaime, Ri, Riri (ree-ree).
Alternatively, I like VolAuVent's suggestions, particularly Winter, Aurora and Stella.
Hope that helps,
MammaT (proud to be a January Girl)