*EDIT: Please see my comments below to clarify what I see as the difference between small-e eve and capital-E Eve.
For what it's worth, I think both answers above (those of Sam K and Tom Cotton respectively) are correct depending on what kind of "night before" you mean.
According to the Oxford English Dictionary, eve is indeed a derivation of evening. However, as Tom Cotton notes, eve is typically used to mean "the day before". This use is apparent quite early, circa 1290. Later (1780), eve was also used to mean "the time immediately before an event". Note that evening nevertheless remained in use to mean that period between dusk and nightfall which is an indication of how eve and evening diverged in meaning.
It's not surprising then that the OED uses eve in exactly this way to define vigil: "the eve of (i.e. preceding) a festival or holy day, as an occasion of devotional watching or religious observance" (OED). This is what is meant by Easter Vigil (as in Sam K's answer), when you stay awake in prayerful contemplation of Christ's resurrection the following day. To that end, Christmas Eve could also be thought of as a sort of vigil, awaiting the birth of Christ. Indeed, attending Midnight Mass on Christmas Eve is surely a vigil in this sense.
The reason I have spent time unpacking the English is that I think eve is a bit of a quirk of English in that it can mean "the time preceding an event" tout court and yet is also applied to events of significance (temporal or religious). But whereas the English can straddle these two uses, my feeling is that Latin differentiates between them.
To this end, I think pridie does mean "eve" in the practical sense of "the time before".
Yet, pervigilium seems more suitable for things like Christmas Eve and even New Year's Eve in that these are a "time of devotional watching" before a festival or holy day. Further, on reflection, I think that pervigilium is preferable to vigilia because Lewis & Short specifically note that pervigilium is "a devotional watching", whereas vigilia is more "a keeping awake for the security of a place".
This difference is apparent in Livy:
... castra Campana ut in pervigilio neglecta simul omnibus portis
... the Campanian camp being unattended because of the vigil he
invaded via all the gates at once
History of Rome, 23.35
Gracchus is able to invade the Campanian's camp at midnight because they were keeping a vigil (pervigilium) before a sacred festival. The terrible irony is that if they had been keeping vigil (vigilia), they would not have been caught unawares!