To find the total surface area of the box, we need to find the area of each face and add them together.
The area of the front and back faces (10 cm by 4 cm) is 10 cm * 4 cm = 40 square centimeters each.
The area of the top and bottom faces (10 cm by 17 cm) is 10 cm * 17 cm = 170 square centimeters each.
The area of the side faces (4 cm by 17 cm) is 4 cm * 17 cm = 68 square centimeters each.
Adding all these together, we get a total surface area of 40 + 40 + 170 + 170 + 68 + 68 = 556 square centimeters.
Therefore, Jerry will need 556 square centimeters of wrapping paper to wrap the present.
So, the correct answer is C. 556 square centimeters.