He may have the perfect name for New York, but few figured he would actually wind up with the Giants.

With their two primary targets swiped off the board just ahead of them — both by teams that traded up to leapfrog the Giants — and two other highly touted players sitting there with off-the-field concerns, the Giants chose cornerback Eli Apple with the 10th overall pick in the NFL Draft on Thursday night.

General manager Jerry Reese did exactly what he said he would do, picking the best player available — but with a caveat.

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“He was the highest-rated player on our board,” Reese said, “beyond the guys with issues.”

Those guys would include linebacker Myles Jack, whose knee injury caused teams to back away from his prodigious talent, and left tackle Laremy Tunsil, who once was projected as the first overall selection but plummeted after video of him smoking from a gas mask bong surfaced on Twitter moments before the draft began.

The players the Giants were believed to covet the most were offensive lineman Jack Conklin and linebacker Leonard Floyd, and it looked as if they might have a chance at them as the top of the draft unfolded. But the Titans traded up to No. 8 to grab Conklin and the Bears jumped up to the ninth spot for Floyd.

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It felt a little bit like all of those heartbreaking games in 2015: So close to victory, only to lose out in the final seconds.

The Giants said they prepared for such a scenario, but if they did, that probably would have been filed under one of the worst cases.

“You don’t get disappointed,” Reese said. “You just stay with your board, and when they come off, they come off. Nobody is crying because somebody is picked. It’s: Who is the next good guy available? And we think we got a really good player.”

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Reese said the Giants considered trading back once Conklin and Floyd were gone but did not get their price. They did not make much of a push to trade up.

“We don’t worry about what other people do,” director of college scouting Marc Ross said. “We were comfortable sitting right there . . . This was a scenario where we were comfortable just sitting right there where we were.”

So they picked Apple, the top cornerback on their board and the second cornerback taken in the draft after Jalen Ramsey went to the Jaguars at No. 5. The Bucs, who traded down from No. 9 to No. 11 with the Bears, took cornerback Vernon Hargreaves III.

Not even Apple himself thought he would be going to the Giants. The native of Voorhees, New Jersey, saw a Garden State area code on his phone and suspected it was one of his friends pranking him. He almost didn’t pick it up. On the other end was Giants coach Ben McAdoo, ready to welcome him to the team.

“It’s a need pick,” Reese said. “It’s a value pick where we had him ranked and it’s absolutely a need pick. If you see our corner depth, you would [know that] . . . When you have two corners in this league, you’re short one.”

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Apple likely will start out playing the slot, with veteran starters Dominique Rodgers-Cromartie and newly signed free agent Janoris Jenkins on the outside.

“I’m excited to get with those guys and get to work and try to be legendary,” Apple said. “That’s the goal.”

He said he has played very little in the slot, other than following opposing top receivers there. The Giants see that position as a starting job, considering how often teams run three-wide receiver sets.

Apple made headlines in recent days when an anonymous scout told the Milwaukee Journal Sentinel that the young player, who won’t turn 21 until August, had no life skills. “Can’t cook” was the almost absurd report, one that clearly did not come from the Giants.

“Cooking is ridiculous,” Ross said, channeling Allen Iverson with a “We’re talking about cooking?”

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Reese echoed that sentiment.

“I don’t care about his cooking,” the general manager said.

What he cares about is that he can help the continuing rebuild of the defense and allow the Giants to move on from what feels like a disappointing night.