My daughter has a permanent residence permit in Italy. Does she need a visa for a 3 month Erasmus program in Norway?
Norway doesn't require a residence permit to be applied for study programs of upto 90 days. Norway is not an EU member state and therefore doesn't have any special provisions for LTRs (long term residents) of other EU member states like most EU countries (Except Ireland and Denmark). But Norway is a part of the Schengen area so an EU LTR can travel to Norway as a visitor for upto 90 days. Prima facie, it seems that your daughter shouldn't require an additional visa for a program of upto 90 days.
Though not frequently enforced in practice, her travel outside Italy in the Schengen region should be within the 90 days in the previous 180 days. So technically, she shouldn't have been outside Italy (in the Schengen area) for 90 days before she begins her travel to Norway and should return to Italy after 90 days. This is of course more of a technicality and will "usually" not pose a problem, especially to those having LTR from an EU country as they have enhanced rights in the EU.
You should still check the UDI website and call them to have your doubts cleared.
Since the application seems to be made through the university, it would best for your daughter to ask the university about the exact conditions of participation.
This page implies that would be up to the university (participating organisation) to ensure that any required visa for Norway is obtained.
Requirements regarding visa and residence permits
Participants in Erasmus+ projects may need to obtain a visa for staying abroad in the EU Member States and third countries associated to the Programme or third country not associated to the Programme hosting the activity. It is a responsibility of all the participating organisations to ensure that the authorisations required (short or long-term stay visas or residence permits) are in order before the planned activity takes place. It is strongly recommended that the authorisations are requested from the competent authorities well in advance, since the process may take several weeks.