Translation:Finally he remembered that today is his wife's birthday.
That's an excellent question. Semantically it's obvious that nifal should be used here, but when I thought about it in more depth I couldn't really put my finger on why. The best explanation I can deliver at this stage is that the action of recalling something spontaneously (as in something popped up) is using nifal, while paal (זכר, זוכר) should be used when he had and retained this knowledge throughout. You can definitely say הוא זכר שהיום יום ההולדת של אשתו, but using סוף סוף indicates there is some action which happened immediately after this adverb, meaning, his recollection of his wife's birthday. Is it any clearer now?
For the first part of the sentence I wrote 'He finally remembered ...' simply because that word order sounded natural to me in English, and was marked wrong. There's nothing wrong in English with the word order in the given answer ('Finally he remembered ...) but my answer should be accepted as well.
If you mean that the last part of the sentence can also be validly translated into English as 'that today was his wife's birthday' then I completely agree with you. Although there are contexts in which 'is his wife's birthday' is correct, the past tense would be more generally used and is grammatically correct.
As it was stated a few times before, when translating nif'al into English, it doesn't necessarily have to be passive in English, too. This would be one of those cases. Or נכנס - entered. You can't translate it as passive in English, because it's not passive even in Hebrew, even though it's in nif'al.
As Mazzorano explained, nif'al נזכר talks about something coming to one's mind, not about being reminded. For reminding, Hebrew uses hif'il (binyan often expressing causative action): הזכיר - he reminded (someone) (or if you will: he caused (someone) to remember), and its passive counterpart huf'al הוזכר - he was reminded.
I believe there is a grey area here. Since there is no corresponding "structure" in my native language and English, nifal is confusing for me. While the teachers are insistently stressing that nif'al is not passive voice of Paal (Qal), it is easier for me to learn it this way (although GRAMATTICALLY wrong it is SEMANTICALLY correct for me). In the Biblical Hebrew lectures by Michael Carasic, he says there was probably a passive voice for qal (paal) in olden days but now it is gone. And he lists nif'al as SOMEWHAT a reflexive structure, like hitpael. My two cents though!...
Deep analysis. Well, somehow, his recollection of his wife's birthday happened on its own, without effort on his part. Ds brought up that the thought popped into his mind. Maybe he was walking around the house and happened to notice a trigger: the gift he got her last year, a birthday text message from someone on her phone, etc... In this case, הזכיר works if you say המתנה משנה שעברה הזכירה לו The gift from last year reminded him. This is strong causitive. The passive nifal would be בגלל המתנה משנה שעברה הוא נזכר ביום ההולדת של אישתו Because of the gift from last year, he was reminded of his wife's birthday. The ב is the preposition, in this case.