What does a "wage" mean in the time of 'The Odyssey'?

The quote is from Odyssey 10.84-6:

ἔνθα κ᾿ ἄυπνος ἀνὴρ δοιοὺς ἐξήρατο μισθούς,
τὸν μὲν βουκολέων, τὸν δ᾿ ἄργυφα μῆλα νομεύων·
ἐγγὺς γὰρ νυκτός τε καὶ ἤματός εἰσι κέλευθοι.

where the Greek word rendered as "wage" is μισθός/misthos, which did denote a range of meanings depending on historical periods and geographic regions; but in its most common context, μισθός refers to recompense or remuneration paid for work done in a limited period. The related word μισθωτός (pl. μισθωτοί) is often translated as day-labourers, wage-earners, or, more commonly in a military context, mercenaries.

In Archaic and early Classical periods, lower-class peasants, known as the θῆτες/thetes, often hired themselves to aristocrats or more prosperous farmers as agricultural servants. They would receive accommodation and daily provisions from their employers and live on their estates as domestics. Older scholarship occasionally equated the thetes with slaves (δουλοῖ), but apart from the risk of falling into debt slavery (which is itself highly dependent on what period and region we are talking about), there is no evidence suggesting they were treated as such. There's another collective term, πελάται/pelatai (pl. πελάτης), meaning literally "one who is dependent / seeks protection," that became almost synonymous with thetes by the 4th century BCE.

Now to answer some of OP's questions:

What sort of wage are we talking about? Certainly in the time when the Odyssey was composed (c. late 8th century BCE), monetary payment in the form of hard specie was not yet in practice. As recompense for their work, a day-labourer would receive non-monetary allowances in the form of wheat, wine, oil, clothing, or a portion of the harvest. Even after cash payments became widespread, non-monetary provisions remained an important supplement of misthos. Besides, as already mentioned, these wage workers were usually provided with accommodation by their employers for the duration of their contract (often verbally agreed upon in advance). Also worth mentioning is the fact that in early times recompense was not paid according to unit of time worked, but predicated on whether specific outcomes outlined in the verbal contract had been fulfilled.

Concerning the "double wage" bit, we'd have to consider the quote in its context. In Book 10 Odysseus encountered the Laestrygonians, who worked day and night alike and called upon each other as they drove their flock in and out. The Loeb reader (1919) gave the following footnote:

_{The meaning appears to be that the interval between nightfall and daybreak is so short that a herdsman returning from his day’s task meets his fellow already driving his flock forth for the following day. Thus a man who could do without sleep could earn a double wage. The passage is plainly due to some vague knowledge of the land of the midnight sun. M.}

What exactly did the author have in mind when he used the word misthos? A little Begriffsgeschichte may help: the word μισθός had its root in PIE *misdʰós, meaning "reward, payment"; it is further traceable to *mey- which means "to exchange," and that does sum up the concept rather succinctly. As The New Pauly puts:

Similar to the Roman locatio conductio, the Greek misthosis comprises a series of remunerated transactions in which one person transfers things (or a person) to another person for use, so that a particular outcome is achieved, or commits themselves to providing labour or a service.

Thus in its most basic (and probably original) sense misthos was simply rewards for the exchange of either things or labour, and that as I see it is what Archaic authors had in mind when they put down that word.