**Explanation:**

To determine the unit digit of the cube root of a number that ends with 3, we need to understand the patterns of the unit digits of cubes.

**Pattern of unit digits of cubes:**

- The unit digits of cubes follow a specific pattern.
- The cubes of numbers 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 have the unit digits 0, 1, 8, 7, 4, 5, 6, 3, 2, and 9, respectively.

**Finding the unit digit of the cube root:**

- Let's consider a number that ends with 3, such as 3, 13, 23, 33, 43, etc.
- We can observe that the unit digit of the cube of these numbers is always 7.
- For example, the cube of 3 is 27, the cube of 13 is 2197, the cube of 23 is 12167, and so on.
- Therefore, the unit digit of the cube root of a number that ends with 3 will be 7.

**Example:**

Let's take the number 63, which ends with 3.

- The cube of 63 is 250047.
- The cube root of 250047 is approximately 63.
- The unit digit of 63 is 7.

Hence, the unit digit of the cube root of a number that ends with 3 will always be 7.

**Conclusion:**

Therefore, the correct answer is option B) 7.