[Gwaihir the Windlord]

The South generally means either Harad, or more often, the South-Kingdom of the Numenoreans -- Gondor. Consider the context and you'll be able to tell which is which.

Aragorn was the heir of Isildur. His family did not become king of Gondor because the claim of Arvedui -- the last king of Arnor -- to the right of kingship over both North and South-kingdoms was denied. The line of the kings in Gondor had been slightly broken. Instead the crown was given over to Earnur, a relative of the king. Thus, Arvedui's missed out on gaining the High-kingship that Aragorn later attained in his place. Arnor was destroyed in Arvedui's reign, so when the line of Anarien finally came to an end in Gondor a few years later, it was too late to give power to a king of Arnor.
However, the son of Arvedui survived in Lindon. The remnant of the Dunedain of the North became the Rangers, and the heir of Arvedui -- who was the only direct heir of Isildur, and the heir of Elendil as well -- became their chieftans. The line was carried unbroken until the birth of Aragorn, who it was seen was destined to rise up and take the kingship again, or else there was no hope in the revival and succour of the still strong but flagging Dunedain, Gondor and Arnor.

It is quite possible that the Stewards guessed at the existence of a Northern heir of Elendil, and yet did not feel it appropiate to yield the kingship to him. It would really have been impossible, I think. The Stewards did a good job anyway, and the Chieftans were kept, waiting, in the North. They played an important role as well. At least, Denethor seemed to be in knowledge of Aragorn's existence:

And you would supplant me with... a wandering vagabond of the ragged house of Isildur?

If it was known that the house of Isildur was 'ragged', then it is likely that secret suspicion of it's existence was also harboured by the Stewards. Another quote:

Some said that he [Denethor] may have guessed Thorongil's identity in truth...

He would have kept it to himself though.

Although it is true that little knowledge of the currect state of Eriador lay in the knowledge of Gondor, as is shown by Boromire's ignorance of 'Imladris'.