I am not a lawyer, but this post represents my understanding of the law. And on top of that I'm an American, so what I do know of law may be inapplicable to UK law. That said:
The "promisor" is the party who has promised to do something, to pay money or perform a service or whatever. The "promisee" is the person to whom that promise was made, who expect to receive something.
"Consideration" is something of value that is part of the agreement. In U.S. contract law, a contract is not enforceable if there is no consideration. That is, a contract is not enforceable unless both parties are giving something. If Al offers to sell Bob his car for $5,000 and Bob agrees, they have a contract. If the next day Al decides the car is worth more than that and demands $6,000 before he hands it over, Bob could take him to court for breach of contract. But if Al tells Bob that he's such a good friend and he likes him so much that he will give him his car, without asking Bob for anything in return, then there is no "consideration" and this is not a valid contract. Bob can't sue Al for not following through on the gift.
"Unilateral" means action taken by one party without the consent of the other. Suppose Al agrees to sell his car to Bob for $5,000, to be delivered in 6 months. During that 6 months Al has a new, expensive stereo installed in the car. When the 6 months are up he tells Bob that he must pay him $6,000 because the car is worth more now with the new stereo. If Bob did not agree to having the new stereo and to paying for it, then this change is "unilateral" on Al's part.
So getting to this specific example. I read a quick summary of the Stilk case. The gist of it was that sailors were promised a certain amount of money for being part of the crew of a ship. Then halfway through the voyage two sailors deserted, so the others had to do their own jobs plus the work of the two missing men. The captain promised them extra money for the extra work. But then when they got home, he refused to pay the extra money. A court ruled that he was not obligated to do so, because he was not responsible for the desertions, and the sailors' contract said that they would do whatever work was necessary for the voyage to be a success.
So the idea is: The promisee's position changed when the other 2 sailors deserted, so they believed themselves entitled to greater compensation. But the court ruled that the change in circumstance was not the responsibility of the promisor, so he could not be obliged to pay them more. The original contract did not say that sailors would be paid more if other sailors deserted, died, became incapacitated, or whatever. Someone cannot demand that the terms of a contract change because of circumstances outside the other person's control. "The slightest detrimental variation in the promisee's position could be seized upon as consideration". The promisee is saying that any negative change in his circumstances is additional consideration that he is paying. The fact that another employee quit means that he now has to do more work, so he is giving more, so the employer should also have to give more. He is "seizing" on the change of circumstance and calling it "consideration".
I think "unilateral" is the wrong word here, as the sailors didn't make their co-workers desert. They weren't responsible for the change either. Perhaps the original author is tacitly admitting that this isn't quite the right word when he says that "in substance" it was unilateral.
Personally, I think the validity of the argument relies on the nature and magnitude of the change. If you promise to sell me your car for a certain price, and then your car is destroyed in an accident, even if you are not responsible for the accident, I don't think you can fairly demand that I buy the wrecked car at the originally-agreed price. But if it got a small scratch, could I use that to call off the deal or demand a lower price? But whatever, the point is not to debate the author's position but to clarify it.