Why did the Institute pick Shaun?

Solution 1:

This is explained in the game - they were after an individual who hadn't already been exposed to the radiation of the wastelands. This is covered early on during conversations with Shaun after arriving at the Institute, when you get the "But Why?" dialogue option.

Father: ...You believed that ten years had passed. Is it really so hard to accept that it was not ten, buy sixty years? That is the reality. And here I am. Raised by the Institute, and now its leader.

Select "But Why?" dialogue choice

You: But why? Why take a child? Why take you?

Father: Ah, now that's the question isn't it? "Why me?". At that time, the year 2227, the Institute had made great strides in synth production. But it was never enough. Scientific curiosity, and the goal of perfection, drove them ever onward. What they wanted was... the perfect machine. So they followed the best example thus far - the human being. Walking, talking, fully articulate... capable of anything.

Select "They needed specimens" dialogue choice

You: So the weird science experiments needed specimens. That's why they took you.

Father: In a manner of speaking, yes. The Institute endeavored to create synthetic organics. The most logical starting point, of course, was human DNA. Plenty of that was available, of course, but it had all become corrupted... in this... wasteland... radiation affected everyone. Even in their attempts to shield themselves from the world above, members of the Institute had been exposed. Another source was necessary. But then the Institute found me, after discovering records from Vault 111. An infant, frozen in time, protected from the radiation-induced mutations that had crept into every other human cell in the Commonwealth. I was exactly what they needed. And so it was my DNA that became the basis of the synthetic organics used to created every human-like synth you see today.