Sally is thinking of a positive four-digit integer. When she divides it by any one-digit integer greater than $1$, the remainder is $1$. How many possible values are there for Sally's four-digit number?
1. **Identify the condition for the number:**
   Sally's number, \( N \), when divided by any one-digit integer greater than 1, leaves a remainder of 1. This means:
   \[
   N \equiv 1 \pmod{2}, \quad N \equiv 1 \pmod{3}, \quad N \equiv 1 \pmod{4}, \quad N \equiv 1 \pmod{5}, \quad N \equiv 1 \pmod{6}, \quad N \equiv 1 \pmod{7}, \quad N \equiv 1 \pmod{8}, \quad N \equiv 1 \pmod{9}
   \]

2. **Find the least common multiple (LCM):**
   To satisfy all these congruences, \( N-1 \) must be a multiple of the least common multiple (LCM) of the numbers 2, 3, 4, 5, 6, 7, 8, and 9. We calculate the LCM as follows:
   \[
   \text{LCM}(2, 3, 4, 5, 6, 7, 8, 9)
   \]
   Breaking down each number into its prime factors:
   \[
   2 = 2, \quad 3 = 3, \quad 4 = 2^2, \quad 5 = 5, \quad 6 = 2 \times 3, \quad 7 = 7, \quad 8 = 2^3, \quad 9 = 3^2
   \]
   The LCM is found by taking the highest power of each prime that appears:
   \[
   \text{LCM} = 2^3 \times 3^2 \times 5 \times 7 = 8 \times 9 \times 5 \times 7 = 2520
   \]

3. **Formulate the general solution:**
   Since \( N \equiv 1 \pmod{2520} \), we can write:
   \[
   N = 2520k + 1
   \]
   where \( k \) is an integer.

4. **Determine the range for four-digit numbers:**
   Four-digit numbers range from 1000 to 9999. We need to find the values of \( k \) such that:
   \[
   1000 \leq 2520k + 1 \leq 9999
   \]
   Solving for \( k \):
   \[
   999 \leq 2520k \leq 9998
   \]
   \[
   \frac{999}{2520} \leq k \leq \frac{9998}{2520}
   \]
   \[
   0.396 \leq k \leq 3.968
   \]
   Since \( k \) must be an integer, the possible values for \( k \) are 1, 2, and 3.

5. **Calculate the corresponding values of \( N \):**
   \[
   k = 1 \implies N = 2520 \times 1 + 1 = 2521
   \]
   \[
   k = 2 \implies N = 2520 \times 2 + 1 = 5041
   \]
   \[
   k = 3 \implies N = 2520 \times 3 + 1 = 7561
   \]

Thus, there are 3 possible values for Sally's four-digit number.

The final answer is \(\boxed{3}\)