What a nice surprise! Looks like the roots are still alive and it put a shoot. What kind of hydrangea is it supposed to be?

Other than adding some mulch, giving it the same care it had before (ask your mother how often she watered it, how often she fertilized it, what fertilizer she used and how much she used), I would do nothing else.

Bring it inside when it goes dormant and the leaves dry out. Once in a protected garage or shed, water it once every two weeks or so. Take it outside when Spring temps arrive.

If you insert a finger into the soil to a depth of 3" or and it feels dry or almost dry, water it. I normally water a little bit, wait for the water to drain and water some more.

Keep it in a location where it gets morning sun, afternoon shade and some protection from wind if the area is windy (the wind can dry out the soil faster than if placed in non-windy areas).

Potted plants need water more often. And all the watering makes the nutrients leach out so they need to be fertilized more often too. During the growing season, I would fertilize but during the winter months I would not. If you water and fertilize as often as she used to when she was in the house, you could give her the plant for Mother's Day or some other special occasion. But if you are panicky of killing it, just drive over to her now, give it to her now and let her take care of it as she used to do!!! Ha!

Your Mother is blessed to have such a thoughtful daughter. I know she will be happy with your efforts even if the hydrangea does not survive. I believe deep in my heart that it will.

you should also not add any fert, can you see any buds at the base of the leaves? hopefully you put it inside so it can grow better and harden off

Yep, store in a sheltered location in winter and keep the soil somewhat moist.

Also, for a hydrangea that small, you should keep it growing in that size pot for approximately 3 years before planting in the ground. This will help it achieve size before it's subjected to stresses of winter. Until planted, the pot should be put outside after last frost and as Luis said, taken back to shelter right after the leaves fall off in the Fall. When outside, keep the pot in a semi-shaded area (50/50) and keep the soil moist.