According to the USDA, if ice cream has been completely thawed, you cannot safely refreeze it. Ice cream is unsafe to eat after it has thawed, and partially thawing ice cream and then refreezing it can affect the quality of the product.
Many foods can be refrozen after thawing, provided there are still ice crystals in the food and it has a temperature of 40 degrees Fahrenheit or less. Following a power outage, check that the internal temperature of the freezer is 40 degrees or less before deciding which foods to keep or discard. If any raw meat juices have come into contact with food, discard that food immediately. Never taste food to determine if it is still good.