I'm not sure about the grammar rule, but it reminds me of early modern English - "for to ___" - that may be found in literature from the 16th-19th centuries. It sounds awkward now, though. Remember though that å huske means "to remember" in the infinitive sense - "to" not pointing to the purpose of having the book, just to indicate the infinitive tense of the following verb. Therefore, to introduce the purpose, the word "for" is used. We no longer do this in English but many other languages do. You could also think of it as "for (the purpose of) remembering."
Sorry of that's not entirely coherent.
The obligation is to make sense AND provide a natural English equivalent. (If you're going to require that every word be accounted for in translation, then we'd better write "She has a book for to remember the day" in English)
I am suggesting English sentences that are more natural than the one provided as 'correct', at least in the dialect of English that I speak. I think the sentence "She has a book to remember the day" sounds awkward, and one I that would never say.
I think this has more to do with how languages work in our heads. I think we can both imagine instances of phrases that are grammatically correct enough in both languages but arnt really said because it would be awkward.
On that comes to mind to me for Norwegian to English is "All of these kinds of things". Its not wrong and makes perfect sense, just weird to say/hear.