You first ask about the meaning of a word (‘sign’), then you go on to ask about the possible meaning of the sign. So, the word itself has a simple meaning, and the context of chapter 36 shows what it is.
Through the prophet Isaiah, God had foretold the fall of Assyrian king Sennacherib who had heaped ridicule on the God of Israel. Chapter 37 starts with the king of Judah, Hezekiah, tearing his clothes in anguish at that blasphemy, and going into God’s temple to pray about it. Then Isaiah is given a message from God, first, that Sennacherib’s threats would come to naught. This was fulfilled by a report coming to Sennacherib that Tirhakah (the Cushite king of Egypt) had marched out to fight him, so he had to about-turn to deal with that. Of course, he intended to return to crush Jerusalem after dealing with this urgent threat to himself. Hezekiah again went to the temple, to lay down this second written threat before God in prayer, seeking deliverance. Isaiah gave him another message from God about God’s predicted fall of insolent Sennacherib. Then comes “the sign” to Hezekiah.
The sign is an assurance of the people being able to eat food from the land, which was meant to assure them that, “Once more a remnant of the house of Judah will take root below and bear fruit above… (37:32-32).” The literal sign (or assurance) that God would deliver the people and cause them to ‘grow’ as a nation would be in the land providing them food in exceptionally difficult circumstances. The sign was a physically discerned agricultural series of event over three years. As these events happened, the people would grow in confidence (faith) that Sennacherib would be crushed by God and they would be triumphant.
What did this sign mean? Here I quote from the New Living Translation Study Bible notes:
“Because of the Assyrian siege and its destruction of agriculture, the
people of Judah would not be able to plant and harvest as usual. The
promise that life would resume after the siege assured them that God
was with them and would provide as they carried out their everyday
activities. They needed to develop their trust in God over a period of
three calendar years in the confident expectation (waiting, 30:15,
18) that God’s word would be true. Perhaps the point was that after
rescue from the Assyrians, it would be too late in the present year
for fall planting. They would have to wait until fall in the second
year to plant again, and they would reap their first crop in the
spring and summer of the third year.” (page 1161)
So, it was less a case of not needing to work the fields, as being unable to. But – not to worry – God would provide food for them until they could reap the results of their work during the second year in the spring of the third year.