What is the significance of the characters's names?

There are quite a few characters in this story that don't have names, so I am going to direct my answer to three main characters that do have names.
Pelayo, Elisenda, and Father Gonzaga are the names of three characters that might be based on historically significant people in Christian history. Pelayo is credited with starting the Christian reconquest of the Iberian peninsula from the Moors. Elisenda could refer to Elisenda of Montcada. She was the last wife of James II of Aragon. She was supposedly quite pious and was noted for her "exquisite religiosity." She was also instrumental in the construction of a monastery called the Order of the Poor Clares in Barcelona. Father Gonzaga's name might be referencing Saint Aloysius de Gonzaga. He was an Italian aristocrat that became a Jesuit. He was even granted an audience with the Pope at one point.
All three of these real historical figures are noted for their faith in Christianity. Angels and miracles would very much be believed by these people; however, as characters in the story, they are very much averse to believing the old man's authenticity as a potential angel. Even if the old man isn't an angel, those three characters do not act in a Christ-like manner. They barely care for the old man, and their main goal is his exploitation for monetary purposes. It's possible that this is intentional anti-Christian biasing on the author's part, but only the author himself could confirm that.

The significance of names in Gabriel García Márquez’s “A Very Old Man with Enormous Wings” serves several purposes.
First, several major characters remain nameless in the story. These include the boy child, the titular angel-person, the neighbor woman, and the spider woman, who is referred to only by her physical appearance. This gives the story a folktale, universal quality that makes it seem like it could be set in any time or place. In addition, the anonymity could indicate a lack of importance about these characters’ individual identity. Although they play significant roles in the story, their personalities are either enigmatic or one-dimensional. Much like the stock characters common in folktales, they are simultaneously important and irrelevant.
This serves in sharp contrast to the three named characters in the story. Pelayo and Elisenda both have names with Germanic origins, although this could be insignificant. The name Elisenda roughly means someone who walks through temples, which has a heavy religious connotation. This is ironic because Elisenda is not a devout religious believer, as her name suggests; in fact, she is the mastermind behind the money-making scheme that profits from the supposed angel. Father Gonzaga, as an official representative of the church, is the most skeptical of the angel’s authenticity, even going so far as to write to the Pope. His inability to believe goes against his role as a spiritual leader in the community. In these ways, García Márquez might be suggesting that supposedly religious people in the story are actually not very spiritual at all.