We are just coming out of a blackberry winter in mid-Missouri. The expression is an Ozark term for the last cold snap of the season. According to lore, if there is thunder in February there will be a blackberry winter in May, usually past the last frost date of May 10.
When combined with a rainy spring, a blackberry winter makes it tough for plants to get a good start on the season. Bees also struggle because their source of pollen is washed way.
There are other expressions for this weather phenomena including "redbud winter" and "dogwood winter," associated with what is blooming. Of the four seasons, spring has the most storm names associated with plants growing.