I can discuss the ways in which this problem played out in its very earliest days, when what we now know as "Antarctica" was merely an imagined continent known as the ‘Terra Australis Incognita’ – ‘the unknown South-Land’.

Until the 17th century, the very existence of this continent was based more on supposition than on fact. Early geographers, most influentially the Græco-Egyptian Ptolemy, writing in 140AD, had imagined a world divided into four gigantic continents. Europe, and what was known of Africa and Asia, was believed to occupy the north-east portion of the globe. This massive land mass seemed to require a counter-balance. From the earliest days, therefore, globes and world maps showed a giant continent south of the equator, girdling the earth and in many cases joining South America and Africa to China.

As the Portuguese and Spaniards pressed southwards in the fifteenth and sixteenth centuries, it gradually became apparent that the South-Land could not be as big as it had been supposed. Ships rounded the Cape of Good Hope and Cape Horn without sighting it, and sailed north-west across the Pacific and east through the Indian Ocean without finding any trace of the mysterious continent. By 1600, almost the only place left to look was in the great blank that still lay south of the Indies and west of the Americas. 

Contemporary charts continued to indicate the presence of Terra Australis in this area. Over the years, elements of fantasy had crept into descriptions of the South-Land, and in the sixteenth century faulty interpretation of the works of Marco Polo led to the addition of three non-existent provinces to maps of the southern continent. The most important of the three was Beach, which appeared on many charts with the alluring label provincia aurifera, ‘gold-bearing land’; sailors often referred to the whole South-Land by this name. The other imaginary provinces were Maletur (scatens aromatibus, a region overflowing with spices) and Lucach, which was said as late as 1601 to have received an embassy from Java. The existence of these provinces was an article of faith for most Europeans; in 1545, anticipating an eventual conquest, the Spaniards actually appointed a governor of Beach – a certain Pedro Sancho de la Hoz, who was one of the conquistadors of Chile. 

In this sense, then, if in no other, Antarctica – the South Land – was fully expected to be inhabited, and indeed to be one of the richest portions of the world.

Sources

Miriam Estensen, Discovery: the Quest for the Great South Land (Sydney, 1998)

Günter Schilder, Australia Unveiled: the Share of Dutch Navigators in the Discovery of Australia (Amsterdam, 1976)

Something that's also worth noting in addition to the answer that /u/mikedash provides is that wackily enough, if your definition of "early" is prior to about 1885 there was a surprisingly widespread belief in the Open Polar Seas, which claimed that the closer you got to the poles, the more temperate the water got. There was supposedly a polar barrier of ice and then somehow warm water currents from the more temperate regions went below the ice and the sun melted things enough to make a northern and southern paradise. From John Wright's 1953's Open Polar Sea:

How did the advocates of an Open Polar Sea explain why there is presumably a milder climate in the vicinity of the North Pole? Before the nineteenth century they usually attributed it to the direct effects of the sun's rays. Plancius suggested that near the Pole the sun shines continuously for five months each year, and that, "although his rays are weak, yet on account of the long time they continue, they have sufficient strength to warm the ground, to render it temperate, to accommodate it for the habitation of men, and to produce, grass for the nourishment of animals."

This reasoning was often repeated, in spite of two fairly obvious objections: the "weakness" of the insulation during the months when the sun shines uninterruptedly, and the fact that the sun does not shine at all during a comparable period each year. Might not these outbalance the warmth received during the months of continuous daylight? Even as late as the year 1869 Professor Maury did not think so, for he wrote: "FROM THE SUN ALONE, FOR SIX MONTHS IN THE YEAR, WE HAVE FORTY DEGREES OF HEAT AT THE POLE! Less than three fourths of this amount would liquefy and open the space around the Pole, supposing it locked in ice."

Hamilton Sides was the first in decades to research this folly (that had actually been supported by a decent number of scientists, and was advocated for in some fairly well respected journals of the time) when he started looking into the disastrous voyage of the USS Jeanette, which he writes compellingly about in his In the Kingdom of the Ice. That disastrous journey ended the theory for all time when the ship got ruinously stuck north of the Arctic Circle for over a year, although prior to it one thing he notes is that one of the wealthy supporters of the voyage to the far north suggested that Greeks and Spaniards be used as crew quite possibly because "they were more used to the hot conditions" expected upon arrival. To your followup question, though, I didn't find anything regarding first contact items brought with the crew of that or any other voyages.

So while the majority of the great exploration voyages to the Antarctic occurred after the Jeanette survivors returned to rather directly debunk the Open Polar Seas theory - the Arctic journeys of the 19th century came first since the US and European powers undertaking them were closer - prior to that, it was widely expected that there were quite possibly unknown civilizations and resources to be explored and exploited there, but no one pushed truly deep into the permafrost. However, most of the earlier adventurers were happy enough to find the several whaling and sealing islands close to Antarctica, since those were enough by themselves to make them rich.