It is the tradition of Greek writers to describe Arabia as wherever there are Arabs.^1 There were multiple Arabias, if that makes any sense. So it is easy to get the impression that the Arabs were a massive and unified entity from reading Greek works.

However, this is not the case at all. Arabia was actually a very complicated place at this time. Several languages were spoken across the peninsula; Aramaic, Old North Arabian, Old South Arabian and to a lesser extent, Greek was sometimes used in the far north. ^2

The Persian empire definitely had contact with the Arabs, as did the Assyrians before them and the Macedonians. ^3

The Arabians were not a unified entity, rather a collection of nomadic tribes, city-states and kingdoms. If Herodotus talks of Arabia or Arabs staying independent from Persia, which I can't say I recall him doing, he is probably talking about a very specific location.

1: Arabians, Arabias, and the Greeks: Contact and Perceptions, M.C.A. Macdonald

2: Greek Administrative Loanwords in Nabataean Inscriptions, Juan Pedro Monferrer-Sala

3: Early Assyrian Contacts with Arabs and the Impact on Levantine Vassal Tribute, Ryan Byrne

Recommended reading:

The Nabataean Army and the Cohortes Ulpiae Petraeorum, David F. Graf

The Arabs in Antiquity, Jan Retso

Arabs, Arabias, and Arabic before Late Antiquity, Michael C A Macdonald

Arabia and the Arabs: From the Bronze Age to the Coming of Islam, Robert G. Hoyland

I'm afraid I can't read Greek and haven't read the Histories, but to my knowledge the Arabian peninsula as we know it was not united until Muhammad. Before then the only area of Arabia that came close to being a proper state were the South Arabian kingdoms based in Yemen, but with various confederations and tribal hegemonies ruling the Hejaz, Eastern Arabia or Oman, not to mention wider north-west Arabia around Jordan. Around the time of Cambyses II you had such polities as the Qedarites, the Minaeans and so on. But depending on what term he is using in the original Greek and the context, it's possible (I would even say 'likely' given the diversity of meanings to terms like 'Saracen' in use in ancient sources) that he's not referring Arabia in the sense that we would use it.

Perhaps more problematically, our written sources for pre-Islamic Arabia are pretty awful. Bronze-age Arabia is not a specialty of mine but it's my understanding that sources like Herodotus-- that is, outside sources that nonetheless might plausibly be well-informed-- are our best bet in many cases, though I'm not sure what the specialist view of Herodotus is.

I'm sorry if that's not more specific but as you can imagine this is a sort of inter-disciplinary question and I'd be curious if anyone better read on Herodotus could chime in.