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.
Recall that in the "Tips" for this unit, it says the infinitive can often act as a gerund (fancy name for noun ending in "ing"). So it may help to think of this as "She has a book for remembering the day". If you use the gerund, then "for" sounds more natural in English. Also in Tips, it defines "for å" as "in order to". So you could also think of this as "She has a book in order to remember the day".