Then, things turned surreal when Tony Conigliaro, already an icon with the Boston Red Sox by then, made a cameo at the graduation to give his younger brother an incredible message.
"[Tony] went up on the stage and said, 'I have to make an announcement.' He said, 'My brother was just drafted No. 1 by the Boston Red Sox.' So that's how I found out -- it was kind of unusual," said Billy Conigliaro.
And so it was that Billy Conigliaro became the first-ever draft pick of the Boston Red Sox back in 1965. Conigliaro was taken fifth overall in that draft, a fact that still awes him.
"I think back to being No. 1, that's something," said Conigliaro. "I was just a high school kid. They had Arizona State to choose from, the University of Arizona, USC, all these teams. I was just a little high school kid. I didn't have much power or anything. I just had a lot of potential."
And plenty of name recognition. The late Tony Conigliaro, thanks to his power, charisma and local roots, was one of the most popular players the Red Sox have ever had.
So the Red Sox thought that perhaps Billy C. could demonstrate some of the same prowess. He showed glimpses of it in 1970, clubbing a career-high 18 homers for the Sox in 398 at-bats. But Conigliaro would play just five years in the Major Leagues, the first three of which were with the Red Sox. He retired because of knee woes following the 1973 season. Conigliaro played in 347 games, hitting 40 homers and driving in 128 runs.
In retrospect, Conigliaro, who still lives in the Boston area and makes appearances at Fenway Park, wonders how things might have been different if any other team but the Red Sox selected him.
"I really wanted to play with the Red Sox," Conigliaro said. "If I was drafted by someone else, it really wouldn't have been that much excitement. As it turns out, I might have been better off going with another team. It might have been a little better for me. Tony was always No. 1. That didn't bother me that much, but it's tough to play in your hometown -- people are more critical."
Conigliaro felt as if he was fighting an uphill climb before he even got to Fenway Park.
"Oh, yeah, when I went to my first Minor League team, Waterloo, Iowa, for A-ball, I was only 172 pounds out of Swampscott High," Conigliaro said. "I only played 14 games [a year] in high school. I really didn't have that much experience. [I was] going out there against these guys who had been playing in the Minor Leagues for 10 or 12 years, and they think you only got drafted because of your brother. There was a lot of resentment, because these guys were trying to make it and I got a pretty good bonus, considering. It was tough."
All these years later, Conigliaro has peace with the way everything turned out.
"I think it would have been a little easier for me if I had gone with another team. It was really tough to play in Boston," he said. "The thing that really killed me was when they traded Tony, I made a lot of speeches about how upset I was. That pretty much sealed my fate there."
If baseball was as advanced in medicine back in 1973 as it is now, Billy Conigliaro likely could have played for several more years instead of retiring at the age of 26.
"I had one knee operation in May , and I had to have another one in September," Conigliaro said. "I still should have stuck it out and tried to play a little more. I look back now and wish I would have tried it."
No matter what else happened, being the first draft pick in Red Sox history is something that can never be taken away from Conigliaro.
"To be No. 1, ahead of all the college kids and everything, I was trying to explain it to my wife -- that's pretty amazing when you think back," said Conigliaro. "A lot of times, first-round picks, they don't make the Major Leagues. It's a tough battle, I'll tell you that."
Ian Browne is a reporter for MLB.com. This story was not subject to the approval of Major League Baseball or its clubs.