The word probation is derived from the French probacion meaning proof/test/trial. In a religious context, circa 1350, it meant a demonstration. In 1363, it began to be recognized as a period of trial for an academic fellow. In 1478 it was considered to be part of the novitiate (period where you have entered religious order, but have yet to take vows) in a religious institute, but by 1549 is was also defined as the time preceding the novitiate.
All definitions of probation have always revolved around the examination or testing of a person or thing, the action or process of putting something to test, investigation, experiment. Today, we consider it more as a restricted time period, whereas the meaning in dated context implies it was used with the understanding of probation as a solitary event. Probation was more of an event than a period. Basically, in 1645 you were probed or put to probation, but in 2016 you are placed on probation.
The word “probation” is only used once in the text. I was not lost by it’s novel use, but I did notice the subtle difference. Horatio says “This present object made probation.” (1.1.171)
In this scene, the ghost has disappeared. Horatio makes a comment that he heard the crow of a rooster and the emergence of day sends ghosts scurrying off, and that the event they witnessed proved it true. This is important because it sets the stage for interaction with the ghost to take place at night. This is important for the setting because the night always seems to hold more mystery and possibility than the day does.
I find it interesting that the historical sense of the word probation meant a test, seemingly prior to earning a rewarding status. Today being “placed on probation” is often a test where the goal is for one to avoid losing a certain type of status or avoiding punishment. It seems that in the scene you referenced, the ghost fulfilled his probation of proving its existence, which is the older definition of the word. I wonder how the word ended up evolving to be a bit more negative like it is today.