It sounds like you’re measuring two things:
(1) The mass attached to the spring.
(2) The period of oscillation.
For minimum error:
(1) Use a high quality scales, correctly zeroed before each weighing.
(2) If you are measuring the period by counting oscillations per unit time: choose a longer time period and try to estimate fractional oscillations when time is up.
If you are measuring the period by timing a certain number of complete oscillations: count a larger number of oscillations. Realize that hand-triggering a stopwatch is only accurate to maybe 100 msec or more, requiring more sophisticated means for greater accuracy.