"It rains at night."
Translation:Il pleut la nuit.
"Strawberry month" uses the noun "strawberry" as an adjective and this is why it is in the singular. There is not a corresponding rule in French were nouns cannot be used as adjectives anyway.
To express repeated times, you will use singular nouns: "le matin, l'après-midi, le soir, le jour, la nuit, le mardi".
All of these mean "every morning/afternoon/evening/day/night/Tuesday".
"Les nuits" is possible only within a limited period of time, like "les nuits d'été" or "les nuits de janvier", where these nights are specific.