I set up a poll on the Esperanto-speaking forum, and to date the results are:
Silvestra vespero 31% [13]
Antaŭnovjara vespero 39% [16]
Novjariĝa vespero 14% [6]
Alia (skribu!) 14% [6]
Number of votes: 36

Of course "new year's eve" refers to the whole day and not to the evening. But who ever attended a new year's eve party that began at ten in the morning? For most people it's a normal work day, except that probably you get off early. At least here in the US. Most shops are open (unless they are very small and the proprietors are on vacation) etc.

So I'm happy to leave "new year's eve day" [if you will] nameless, and only refer to the evening as "new year's eve".

wuerfel:

Chainy:Why not call it 'Novjara Festo'? This is probably quite clear for everyone.

The problem is that the “Novjara Festo” is the 1st of January, a public holiday in many countries. The Day of Pope Saint Silvester, on the other hand, is the 31st of December.

The “novjariĝo” (or “turniĝo”) is what happens at midnight, when the old year ends, and the new begins.

I'm a little confused. The big party (or 'festo') takes place in the evening leading up to midnight. So why not refer to this as the 'Novjara Festo'? (I think we are generally referring to this party when we say 'New Year's Eve')

And the 1st of January could be referred to as 'Novjara Tago' or 'Novjara Ferio' (during which we tend to lie around in bed recovering from the 'festo' of the previous evening)