What did Jesus mean by saying 'to the right of the boat '?

St. Thomas Aquinas explains why He says "right side" (and not "left side") in his commentary on John 21, lecture 1:

2589. Secondly, the Evangelist mentions Christ's order, Cast the net on the right side of the boat. In Luke (5:4) there is a similar incident, but there Christ did not tell them to cast their nets to the right side, as he does here. The reason for this is that the fishing mentioned by John signifies that fishing by which the predestined are taken to eternal life, and it is only those children on the right who are brought there: "The Lord knows the ways that are on the right; those on the left are perverse" [Prv 4:27]; "The right hand of the Lord does valiantly!" (Ps 117:16). The fishing mentioned in Luke signified the call into the Church, and so the net is cast to all sides because people are caught and brought to Christ from all over: "Go out quickly to the streets and lanes of the city, and bring in the poor and maimed and blind and lame" (Lk 14:21).

I don't disagree with the answer from L1R, but I would like to add another viewpoint.

In the first event, as recounted in Luke, Jesus

asked him to put out a little way from the shore

In other words the boat was close to the shore, in shallow water (and obviously unsuitable for fishing). Jesus asks them to take the boat away from the shore, where the water is deep.

Put out into the deep water

In the second case the disciples are already fishing, and therefore already in deep water. There is no need for them to "put out into deep water". But he does tell them exactly where to find the fish:

the right side of the boat

It would make no sense to 'correct' the reading to be "into deep water" in the second case, or to try to make the meaning the same (again, remembering that these are two entirely different occasions being described).

My Parish Priest, who incidentally belongs to a coastal area, has this explanation: An average fishing boat of Jesus'time measured 27 by 7.5 feet , about four feet deep and was large enough to hold about 15 people (Courtesy: TheCompass). The crew would comprise of the rowers and the net-bearers. Most rowers occupied the right side of the boat so as to be able to use the muscle-power of their right hand to the fullest. The net bearers occupied the left side,and threw the net standing backwards. That again, would enable them to use the maximum force of their right hands. The net, once cast , would be dragged along with the moving boat so as to trap maximum fish. Now, casting net on the right side of the boat would call for an exchange of positions, and would also demand exertion on the left hand of both the rowers and the the net-bearers. The direction of Jesus to think out of the box and to go beyond the conventionally accepted methods of local fishermen, was in deed challenging to Peter and his team. That they complied with His direction, shows how much they trusted in the Lord.
PS: I accept that my Parish Priest is not the Catholic Church. But, I am inclined to accept his theory till I hear something authentic from the Church.

I would argue that you are missing the significant point of the story.

See the next verse of the text you mentioned in John 21

And He said to them, “Cast the net on the right side of the boat, and you will find some.” So they cast, and now they were not able to draw it in because of the multitude of fish. 7 Therefore that disciple whom Jesus loved said to Peter, “It is the Lord!” Now when Simon Peter heard that it was the Lord, he put on his outer garment (for he had removed it), and plunged into the sea.

This is Jesus revealing himself to the disciples. This is a callback to when he had original met Peter, which is the other account you had mentioned in (Luke 5)

But Simon answered and said to Him, “Master, we have toiled all night and caught nothing; nevertheless at Your word I will let down the net.” 6 And when they had done this, they caught a great number of fish, and their net was breaking. 7 So they signaled to their partners in the other boat to come and help them. And they came and filled both the boats, so that they began to sink. 8 When Simon Peter saw it, he fell down at Jesus’ knees, saying, “Depart from me, for I am a sinful man, O Lord!”

It's important to note that these are not two different accounts of the same event. The event in Luke happens when Peter and Jesus are first meeting each other, and the event in John happens after Jesus death and resurrection, ~ 3 years later.

So there is no reason why the two phrases "To the right of the boat" would need to match "Put out into deep water". Because they are two different occurences, years apart. However, both with a similar point, revealing the Lord to his disciples, by showing his power over nature.