Upon meeting back up with Nick in his agency, he will sit the Sole Survivor down for a talk about Shaun, as well as the kidnapper.
After giving him the details needed to make an educated analysis of the situation, Nick will then ask for the Sole Survivor's input as to which group would possibly be behind the kidnapping. It is agreed that groups such as the Gunners and local raiders are unlikely to be behind such a covert mission, and the likely suspect is agreed to be the Institute, due to their technology and reputation across the Commonwealth.
Trying to picture the kidnapper using the Sole Survivor's description, he deduces that a man of a similar, if not identical appearance, by the name of Kellogg, owns a house within Diamond City, and was presiding there shortly before the events of Nick's kidnapping.
At this point, it is revealed that Kellogg's house is locked up tight, and cannot be lockpicked without the skill to pick Master locks. The Sole Survivor will then need to travel to Mayor McDonough's office up in the stands, and either steal a copy of the house key from the mayor's desk, persuade the mayor or his secretary Geneva with a moderate Persuasion check, or bribe Geneva with 250 caps. Alternatively, giving Mama Murphy chems will allow the stadium seats beneath Kellogg's house to be examined, which will result in the Sole Survivor obtaining a key.
Once a copy of Kellogg's house key has been obtained, the Sole Survivor and Nick will need to thoroughly explore the abode, in which Nick will comment on the fact that the house appears much smaller on the inside, and that there must be a hidden entrance into a secret portion of the home.
Under a desk in the living area, a button must be pressed, which will open the nearby door and reveal the secret section of the house.
Once these items have been retrieved, and it is revealed that Kellogg nor Shaun are anywhere in the house, Nick will deduce that Dogmeat can be used to track Kellogg's scent through his brand of cigar and ale.
After leaving the house, Dogmeat will be waiting on the two, and the option to bring Nick along will be presented, concluding this portion of the quest line.
Go to Valentine's Detective Agency
Talk to Ellie
Tell Nick Your Story
Get the Key to Kellogg's House
Investigate Kellogg's House
Show Dogmeat the Cigar
I've found Kellogg's favorite brand of cigars - San Francisco Sunlights. Valentine suggests I show the cigar to Dogmeat to see if he can pick up Kellogg's scent.
Once Nick Valentine is met back in Diamond City, he will become a permanent companion.
It is not necessary to have met Dogmeat prior to this quest - Nick Valentine already knows him and will call for him with a high frequency sound.
Completing of the Getting a Clue quest is a condition for the Far Harbor add-on to begin.
↑Mama Murphy: "You're on the trail of the man who took your boy. I see a house. Locked up tight. Chains put on by a smiling shark. But there's another way. I can see it. A key. Lost and forgotten. Fallen down below. Search the chairs beneath the house. They hold the secret." (Mama Murphy's dialogue)