What does tībīcĭnĭum mean in Latin?

tībīcĭnĭum, a noun, means “the art of playing the flute, playing the flute,”.

What is the genitive of tībīcĭnĭum?

The genitive singular is tībīcĭnĭī. tībīcĭnĭum is neuter, second declension.

the art of playing the flute, playing the flute,
Si ex oliva modulate canentes tibiae nascerentur, num dubitares, quin inesset in oliva tibicini quaedam scientia?If melodiously playing flutes were born from the olive tree, would you not doubt that there exists in the olive tree some knowledge of playing the flute?Marcus Tullius Cicero, De Natura Deorum II

tībīcĭnĭum, ii, n. tibicen,

I. a playing upon the pipe or flute, a piping, fluting (class.): tibicinii scientia, Cic. N. D. 2, 8, 22; id. Or. 58, 198; App. Flor. p. 341, 28; 342, 20; Gell. 4, 13, 2.