If a stem tip cutting, wrong time of year. These need to be collected during normal, seasonal growth....so not in midwinter :-) A root cutting is what you need when taken during dormancy but that does not sound like what you have.

I don't have experience with blackberry cuttings; but what you have is a blackberry scion if I compare it to the fruit trees I do have experience with. Fruit wood cuttings are called scions. If I keep a scion in a plastic freezer bag with some damp newspaper or peat moss around the cut end of the cutting they will last out the winter if the cutting or scion is dormant.

Some sources seem to just stuff some of the damp paper or peat in the bag and the moisture seems to keep what's in the bag and not in contact with the cut end alive. You can keep the plastic bag in a refrigerator unless you have fruit like apples in there. So i keep mine in an unheated integral garage and last almost as well as in the frig.

Scions typically refer to grafting material only, not starting a plant from cuttings! For stem cuttings to take root, they generally require plant tissue that is also in active growth or well budded, not dormant. Holding over the cuttings until a suitable time to plant - refrigerated or not - is not going to work in this case.

From my experience scions start to leaf out if the temperature increases in my unheated space. They will leaf out, although they don't green up unless, or until, they are in sunlight, which I try to avoid.

I'm afraid OP's cutting has been disposed of. But if it hasn't sounds to me like there's nothing to loose except a freezer bag and some cutup newsprint. The newsprint or peat should only be slightly damp; not dripping wet.

edit to: "unless, or until"

For the best chance of success, have your friend tip-root one or more of his blackberry canes next autumn and plant those when the ground is not too cold or wet. You will then get good-sized new canes the following year.