Given that student A and student B have the same average score for jumping rope in 1 minute, we compare their score stability by looking at their variances. The variance measures the spread of the scores around the mean; a lower variance indicates that the scores are more tightly clustered around the mean, implying greater stability.

- The variance of student A's 1-minute jumping rope score is given as $S_{A}^{2}=0.06$.
- The variance of student B's 1-minute jumping rope score is given as $S_{B}^{2}=0.35$.

To determine which student's score is more stable, we compare the variances:
$$S_{A}^{2} = 0.06 \lt 0.35 = S_{B}^{2}.$$

Since a lower variance indicates greater stability, and $S_{A}^{2} \lt S_{B}^{2}$, it follows that student A's score is more stable than student B's score.

Therefore, the correct answer is $\boxed{A}$.