Well as nobody anwsered I will give my 2 cents.

The portuguese for one were heavily influenced by Prester John stories, and by all pointed out evidnece, believed him to be real, or at least it was very possible for him to be real. Ofcourse it probably varied by who did you asked, and also when did you ask (becasue Portuguese were going around Africa for a few generations)

But still: The Prester John idea or at least the possiblity was very real in Portugal.

Before Vasco Da Gamas journey, the Portuguese sent two spies (one was Pero da Covilha) through Arab lands into India to gather information about geogrpahy, trade routes, politics etc of India. They also had clear instruction that after the primary task completed, they should seek out the lands of Prester John.

Captains were given instructions to wherever they go, try to find out about Prester John, if anything came up to seek him out and try to establish contact and then an alliance.

And they did, all the way around Africa and in India they asked and looked for clues about any christian kingdoms, followed wild chases based on their wrong interpretion of various information and jumps to conclusion. Apparantly even when Vasco Da Gama finally arrived in India, at first they considered Hindu people Christian, before realizing their mistake. Don't quote me on this one, though

It should not be that strange that Portuguese were so interested in Prester John. Their culture was heavily influenced by "Reconquista" and fighting, especially against the Moors. And a possible alliance with Prester John in their eyes would mean an attack at Muslims from two sides, resulting in total victory, wiping the infidels out completly.

This narration gave a new dimension to the exploration attempt: except of it being adventure, discovery and opportunity&gamble it became a crusade, a possible upmost strategic, military and religious matter. It certainly helped convince a whole new sort of people into the exploration process:military commanders, religious leaders, men with causes and dedicated to faith, men with finances, men wanting to be a part of the bigger picture, and men behind institutions

This was reflected as early portuguese explorations were funded and spearheaded by their monastic knight orders - whose very cause of existence was the fight for Christ. By knowing that, it is definetly understandable that search for Prester John had been used as a reason, or if you more cynical: an excuse, to gather funding, resources, men and the will to set off.

But all that being said, it wouldn't work if the search for Prester John was a sole motivation.

The exploration and drive around Africa was very profitable, and the Portuguese had various direct short and long terms strategic benefits, none of those needing or counting on the Prester John to exist. Finding him would be awesome and possibly game-changing, but in reality his existance did not matter.

Mostly I gathered this from the book A History of Portugal and the Portuguese Empire, Vol. 2 by Disney

p.s. Also if the idea of Prester John and believing in him is amusing to you, I would just point out that right now we have an organisation on this planet, which has been funded generously (in millions) to try to find out if aliens exist and are communicating to us - the SETI program, all while having no real evidence alines exist (other then the educated guess that the universe is too big for us to be alone) and especially no evidence that they are trying to communicate to us. I would say in medival times, they had the same educated guess that prestor john could exist. (not to mention that technically he did exist in the form of Ethiopia, just nowhere near the image Europeans had of him)