In ʿādā ʿidāʾan, *ʿidāʾan* is a verbal noun in the accusative, placed after its related verb. It serves as a cognate accusative (mafʿūl muṭlaq), reinforcing the occurrence or manner of running.
Compare:
ʿĀdā al-farasu — The horse ran.
ʿĀdā al-farasu ʿidāʾan — The horse ran vigorously.
The repetition is therefore not ornamental alone; it gives the action force in sound and meaning.