Hebrew slang is predominantly Palestinian Arabic, or derivations thereof—e.g. mastulמסטול (stoned, zonked), ahabalאהבל (imbecile), dir balakדיר בּאלאכּ (God help you [if you do this]), saḥbakסחבק (close friend), fadiḥahפדיחה (embarrassing mistake, booboo), etc.
Hebrew idioms are predominantly literary and usually of biblical or Talmudic origin—e.g.:
People usually started by simply adopting the new surname informally, often followed by their original surname in parentheses, until their new surname becomes better known among the public—e.g. David Ben-Gurion (né Grin). Then after a year or two, they made it official by registering it at the Ministry of the Interior (or its equivalent during the British Mandate period). This was useful, as in some cases the user might decide to change the name to something else, or to tweak it (as happened in my own family).
It was a kind of nationalist affectation, to proclaim the ancestral, pre-Babylonian-exile, Israelite origins of the newly-independent Hasmonean state—a bit like the motto of the British Royal Family (Dieu et mon droit – ‘God and my right’) is in French, harking back to its Norman origins.
Modern Israel, by the way, does much the same: the Old Hebrew script is used on the modern sheqel coin (bottom left):
The difference is, in Hasmonean times, people knew what the Old Hebrew text said, whereas 99.9% of modern Israelis haven’t a clue: most people assume it says ‘sheqel’, but in fact, it spells Yehud, which ironically is not Hebrew, but the Persian name for its Judean province, dating back to the sixth century BCE, when Persia had just conquered the Babylonian empire, and allowed all exiled nations (the Judeans included) to return to their ancestral homes.