As our wonderful /u/qed1 had explained with examples some time ago, ancient and medieval Europeans were well aware that in the northern latitudes there would be long days and nights, and they came to this conclusion from theoretical knowledge of astronomy and circles alone.

I myself will add quote from the work of Sacrobosco, a 13th century scholar whose work De Spherae was popular among the Portuguese in 1500s so much that it was printed and provided to their navigators and pilots on all their ships.

In the book Sacrobosco says the following for the day length for latitudes above arctic circle:

o those whose zenith is between the Arctic circle and the North Pole, it happens that their horizon will intersect the zodiac in two points equidistant from the beginning of Cancer. And, as the firmament revolves, this intercepted portion of the zodiac always remains above the horizon. Hence it is clear that, so long as the sun is in that portion, there will be continuous day without night. Therefore, if it is as much as one sign, there will be continuous day there for one month without night. If the sun remains for the extent of two signs, there will be continuous day without night for two months, and so on. It likewise happens that the portion intercepted between two points equidistant from the beginning of Capricorn is always left below the horizon. Hence, when the sun is in that intercepted portion, there will be one continuous night without day, short or long according to the extent of the portion intercepted

For the North pole he makes it quite clear:

Wherefore, one half of the whole year will be one artificial day, and the other half one night.

So the fact that there was that there would be 6 months of daylight there would hardly surprise anyone. And in fact, it is the possible implication of this fact that caught many people's attention.

As many men wondered, if there is 6 months of uninterrupted sunlight, wouldn't all the ice melt? And if the ice melted, wouldn't it allow for passage for ships? And if ships could go there, wouldn't it be a much better and shorter way to get to Asia and "Cathay"?

And this is exactly why so many explorers searched for the Northeastern or Northwestern passages. They were all influenced by the hope that so much sunlight would allow for a summer passage through the Arctic.

 

I am only roughly aware of all the expeditions up North but there were many conducted and even more proposed. The idea might have been floating around since the early 1500s. Sebastian Cabot might have been pitching this idea to Venice around 1522, and I think one of the earliest proposals recorded was in 1527 by Englishmen Robert Thorne. In 1550s the English with Sebastian Cabot's support actually attempted to sail this route, but Sir Hugh Willoughby and Richard Chancellor's expedition couldn't get further then northern Russia.

Several expeditions were apparently attempted since then to 1580s, but my knowledge of them is fairly limited. and the peak of the search happened in the 1590s and 1600 with famous William Barentz and Henry Hudson and others, many of them spurred by Petrus Plancius, a dutch clergyman scientist, a loud advocate of the navigable polar sea in the summer.

Not everybody agreed with this line of thinking, and practical experience was showing that the North was cold, even in the summer. And while Plancius reconciled this by proposing there was a passage through wall of ice to a sailable sea, others were more inclined to explain the observable lack of heat despite the constant sun to various other factors. Like the above Sacrobosco who proposes quite erroneously:

As for that argument again, that there ought to be perpetual light there, it should be said that the air there is misty and dense, for the rays of the sun there are feeble and raise more vapors than they can consume, so that the air is not clear.

 

To repeat once more, it was well established that there would be long days and nights in the Polar latitudes. It was also a source of much hope to reach Asia by hoping the daylight would melt the ice, and many men died founding out it wasn't true