What is your interpretation of the relationship between Mary and Billy (Will)?

The relationship between Billy and his mother Mary is a codependent one. By vying for his mother's love and approval in the shadow of his abusive father, Billy gives Mary power she would not have otherwise. When he gets older and grows up, she loses that power and the two eventually have a less personal, less codependent relationship. 
As a child, Billy relies on the love, protection, and acceptance his mother offers him. He is unable to get it from anyone else. As a result, Billy shapes himself into the kind of person she expects throughout his childhood. 
Mary's husband Constantine abused his wife and children, especially Billy. For example, one night when Billy finds his parents filling their Easter baskets, Constantine thinks of his son's small frame and how at five he already "had a scrawny neck and a squeaky, pleading voice." He is annoyed by his son's presence and disappointed in Billy, so he shakes him hard after Mary tries to defuse the situation by telling Billy that it is just a dream. Mary intervenes for him and makes her husband stop.
Constantine decides he has not done enough and slaps Billy's behind hard—making him fall. He goes to hurt him more, and Mary gets in between them. This type of interaction explains the relationship between Billy and Mary during his childhood. Because her husband is abusive toward Billy, Mary is more focused on him; she dotes on him more than his sisters—always in an attempt to stop Constantine from abusing Billy.
Billy wants to please Mary. At one point, she chooses to buy him a stuffed monkey because that is what Mary wants him to prefer. Billy himself wants a Barbie, but he does not tell his mother. Part of the reason he wants to please her is because she is his protector and the one who stands up for the person Billy is inside. His father sees Billy as he wants him to be—a boy who wants to buy a football rather than replace a broken toy horse. Mary accepts the desires that Billy voices and continues to love him, even though that love stifles him.
Billy is appreciative of the love Mary has for him. At the same time, though, it overpowers him and prevents him from coming into his own. Cunningham writes the following:

She smiled, deeply pleased, and told him he was a good little boy, a prize. She told him he deserved all sorts of good things, and if anyone tried to tell him differently, that person didn't know what he was talking about. Billy stared at her gratefully. She offered a practiced smile, one he'd seen thousands of times: a quick jerking upward of the corners of her mouth, a squeezing shut of the eyes, as if the act of smiling caused her a sharp and exquisite pain. Something bucked inside him, a feeling so unruly he thought he might be sick. She was his friend. She was the one who allowed. How could he dislike her? 

Mary appreciates Billy because he is the kind of child she can connect with. Cunningham says that "of Mary's children, only Billy wanted the things she wanted to give." He is happy to do as she asks. He is polite and studious and does not rebel. Billy is the only one of her three children that Mary really understands.
As he gets older, Billy separates himself from his mother and she feels desperate over the loss of the relationship. She does not understand why he might not want to attend Harvard and why he is not lining up for the life she imagined for him. In the same token, Billy sees her as a barrier to having his own life shaped by his own desires. 
When he leaves home, Billy changes his name and goes by Will instead, further separating himself from the child Mary understood so clearly. This puts Mary more in line with her husband, who has never understood Billy. When they visit him at college, the changes are more clear to them. Cunningham says:

Again, her emotions rose in such confusion that she felt the moisture break out along her upper lip. She wanted to defend Billy from his father. She wanted to stand next to Constantine and demand to know who Billy had turned himself into. How had he gotten so lost? Her lungs clenched up and she struggled for a breath.

She sees Billy as a reflection of herself, and, now that he has changed into a more sullen, unkempt person, she does not have that connection anymore. It frightens and upsets Mary and keeps her from being able to connect with the person Will is. She is faced with this when she visits him on the day he is about to graduate from college. It is clear to her that she does not know him. "Although she thought of Billy constantly, she thought of him in faintly abstract terms, the way she'd think of a character in a television show when the show wasn't on."
As Will gets older, she is able to visit with him a couple times every year in Boston. She still thinks of him as Billy, though she calls him Will when she speaks to him. Even though they are together, the relationship between them is more stiff and formal than in the past. 
Mary is aware that her son is gay and is happy not speaking about it. Once Will meets Harry, however, she has to acknowledge and accept it on a more significant level. It is the first time in years that she sees him as the son she once had—recognizing the same pride and embarrassment in his declaration of love for Harry that he had when he brought home a necklace he had made at summer camp as a child. 
Mary recognizes that she can hurt him for the first time in years when he asks if she finds his orientation strange. However, she says she is not shocked, and she asks to know more. It is the right choice for their relationship. Even though she still feels anger for all the things he did not achieve, she still loves him. Her acceptance of him is the first time they have really connected since before he left for college. 
In the end, the codependent relationship between Billy and Mary gave Mary power that she did not have otherwise. Her husband controlled her, and the world she lived in was not the one she wanted. Young Billy worked for her attention and approval until he finally left the family home to grow into an adult. In the end, Cunningham writes,

By accepting him she had lost much of her power, and she saw that she would not be able to get it back even if she wanted to. He’d moved beyond the reach of her disapproval. She had released him.

She did accept Will. Mary recognizes who her son is and is able to love him for that. She also recognizes that Harry is a good man—the kind of man she might have chosen for herself if things were different. This creates a bridge between them that allows them to have a less codependent relationship and develop a more normal one, even if they are not particularly close. Mary still feels more connected to him than to her daughters. She recognizes that she has to love him as he is; there is no other option.