And then, with the 25th overall selection -- six picks before the first round came to an end -- the Angels took a chance on the switch-hitting backstop.
Eun, born and raised in South Korea until immigrating to the U.S. in 1986, came from a culture that preaches studying hard, going to college, obtaining a degree and ultimately working a normal 9-to-5 job.
Draft night brought mixed emotions.
"It was half and half," Eun said of how she felt to watch her son get selected in the first round. "I was happy, but then another side was like, 'Aw, he should go to school.'"
But Eun can laugh about all this now. Her 25-year-old son is living out his dream in the Major Leagues, while serving as a backup catcher on the Angels and doing what only one percent of those who aspire to take on his profession actually accomplish. Better yet, he's playing for a team whose home ballpark resides in Anaheim, a half-hour drive from the Huntington Beach area where Conger grew up.
About four hours before gametime, Conger will call his father to inform him whether or not he's in the lineup. If he is, Yun (employed by Tesoro Wilmington Refinery) and Eun (a bookkeeper at a private practice in El Monte) can easily make the short trek to Angel Stadium, taking their usual seats in Section 210.
"Whenever we want to see Hank play, we can see him," Eun said, her husband by her side. "That's the best part."
The everyday grind of Major League Baseball means Conger doesn't really ever get to spend Mother's Day with his mom.
But that same schedule has given him a deeper appreciation for the holiday.
"Ever since I've been in pro baseball, it's actually been more special, even though I'm not with her, just because I kind of know that I'm away all the time," Conger said. "For me, just the thought of Mother's Day just kind of brings back everything, of really the sacrifices that she really just put through."
Conger was a big kid growing up, so Yun had him play basketball early on. Then, during the summer when he turned 8, he enrolled Conger in baseball just so he could stay active year-round. Two years later, Conger had fallen in love with the sport so much so that he gave up basketball altogether.
"It was a good thing," Yun said, laughing. "He would've never made it as a basketball player."
Conger got good fast, navigating the country on esteemed travel teams and ultimately being named a second team All-American, donning Gatorade Player of the Year honors and garnering a host of other accolades.
His mother didn't know a lick about baseball, and nobody on either side of Conger's family ever really played the sport. But there she was all along -- traveling with him to showcases galore and even shagging fly balls on the days his father threw him batting practice.
Part of Eun is still stunned her son parlayed that into a big league job.
"I never believed it," Eun said. "I knew that he was good, but I thought to be a Major Leaguer and to get drafted in the first round -- that's other people, not my kid, you know what I mean?
"He's been successful. That's good for us and good for him. But the most important thing is he's a good person with a good personality. That's the things that we care about."