لَيْلَةٍ a night in Arabic

وَتَقُولُ هَذَا آخِرُ يَوْمٍ مِنْ شَعْبَانِ وَأَوَّلُ لَيْلَةٍ مِنْ رَمَضَانِ،

And she said, "This is the last day of Sha'ban and the first night of Ramadan."

لَيْلَةٍ night

This noun is the owned half of 'the first of (the) night', so it sits in the genitive that the head word imposes. The two nouns press together with no 'of' between them; the case ending alone tells you which noun owns which, here making the night the period whose first instant is meant.

From A Night of Reckoning

وَكَانَا فِي السِّجْنِ مَعًا، فَمَنَعَهُمَا السِّجَّانُ مِنْ إِدْخَالِ الْحَطَبِ فِي لَيْلَةٍ بَارِدَةٍ،

They were together in prison, and the jailer prevented them from bringing in firewood on a cold night.

لَيْلَةٍ a night

This noun has no al- and carries a doubled-vowel tanwin ending, marking it as indefinite, 'a night', some night. The 'in' before it sets it in the genitive form. Its indefiniteness matters: it presents a non-specific cold night as the setting.

From A Son Protecting His Father

يَفْعَلُ ذَٰلِكَ كُلَّ لَيْلَةٍ مِنْ تِلْكَ اللَّيْلَاتِ الثَّلاَثِ

He does that on each of those three nights.

لَيْلَةٍ night

A singular noun in the owned 'of' slot of 'each ___', so it carries the genitive ending; 'each' plus a singular is the fixed way to mean 'every single night'. It is the second, owned half of that pair.

From The Secret Migration

لَيْلَةُ الْخَنْدَقِ فِي لَيْلَةٍ بَارِدَةٍ

The night of the Trench was cold.

لَيْلَة a night

This is the noun governed by the preposition 'in', which is why it takes the genitive ending. Unlike the earlier 'night', this one is indefinite ('a night', not 'the night'), so it introduces a fresh, unspecified night rather than pointing back to a known one. The adjectives that follow will all describe this noun.

From A Spy in the Enemy Camp