Refers to money one carries for personal expenses; somewhat old-fashioned, often replaced by 小遣い (こづかい) in modern speech.
祖父はいつもふところぜにをいくらか持っていた。
My grandfather always carried some pocket money with him.
Compound of 懐 (ふところ, 'bosom; pocket') and 銭 (ぜに, 'money; coin'), literally 'bosom money'. The exact historical derivation is uncertain.