Born Detective

The culprit must be the one who forgot to change his diaper.

Kid Detective who grows up to be a Private Detective, with the help of his/her Badass Family. In short, a parent was a PI (or other profession) and raised their child from a young age to help Mommy/Daddy spy on other people, etc.

Examples of Born Detective include:

Anime and Manga

Although not a detective himself, Shinichi Kudo's father is a mystery writer, thus giving Shinichi his detective streak and easy access to the books of Conan Doyle. The fact that he and Shinichi's mother are able to put on a performance that baffles even Shinichi early on, they both clearly have the background to qualify Conan for this trope.
- To help further the cause, Yusaku Kudo is Brilliant but Lazy. Megure and Kogoro have both tried on several occasion in his Backstory to recruit him to being a police detective...just for Yusaku to catch on and prove smarter.
Light Yagami's father is a police inspector and it's mentioned that Light has used his vast intellect to help solve crimes in the past. For this reason, he ends up helping the Kira investigation. Too bad he is Kira...

Literature

Isabel and Rae Spellman of the Spellman Files book series. (David appears to have been brought up to be a PI as well, but he rebelled.)
Encyclopedia Brown's father was a police detective and amazingly enough all the cases that stumped his entire police force were always solved by his son over dinner.
Joe & Frank Hardy, who helped their PI father solve cases.
- Or, in some cases (especially where a reward was involved) competed with their PI father.
Sherlock Holmes, in the single story where he talks about his family ("The Adventure of the Greek Interpreter"), acknowledges that his observational skills are a family trait, and that his elder brother Mycroft has an even greater share of them than he himself.
- In the pastiche series, Enola Holmes, the title character, who is the far younger sister of Sherlock and Mycroft, runs away and finds that she is no less talented as a detective as her brothers.
- And the tradition continued into at least the late 20th Century with Shirley, Mycroft's Canadian-born great-granddaughter (Or it might have been one of Holmes' other brothers. Sources vary.).
  - There's a new series called The Sherlock Files which feature the great-great-great grandchildren

Walter "Ramses" Emerson is the son of Amelia Peabody and Radcliffe Emerson, both archeologists and both Amateur Sleuths. He has been detecting with them since he was six or so. Whether they want him to or not. He follows them in both careers, with a side-order of secret agent.

Live Action TV

Veronica Mars, raised by her sheriff-turned-PI dad, was already running her own PI business in High School (it was well-known that the girls loos doubled as 'her office' during school hours). A lawyer-friend of the family would leave cases open on the desk she had at her father's office while he was with clients in case she decided to 'take a look'.
Emerson Cod of Pushing Daisies (who started out so early, he was being used in stings as a baby)
Shawn Spencer of Psych was trained by his police officer father to have observational skills on the Sherlock Holmes level. He uses them to pretend he's psychic.
On The Wire, Detective McNulty uses his preteen sons trail a drug kingpin through the supermarket, and judging by their skill at it, he's taught a few things about tailing a suspect. While he is a loving father, this is an example of how terrible a person McNulty is at everything but being a detective. His ex-wife is pissed when she finds out and tries to take away his custody rights.
He wasn't raised to be a detective, but Monk's neurotic mother probably caused him at least in part to develop the OCD that makes him such a skilled investigator, and the episode "Mr. Monk and Little Monk" revealed that he solved his first (admittedly minor) crime in middle school.
- A mini series called Little Monk was created during its eighth season which shows Adrian and his brother Ambrose serving as his Watson solving several cases in school.
On Crossing Jordan, Jordan Cavanaugh was a crime-solving medical examiner whose father, Max, was a retired cop.
Both Deb and Dexter are cops, as was their father, Harry. Played with, in that Dexter is actually Harry's adopted son (and a serial killer).

Newspaper Comics

Dick Tracy's adopted son Junior accompanies Dick on cases during his childhood and grows up to be, er, a police photographer.

Video Games

Persona 4´s Naoto Shirogane, who is the fifth generation in a family of professional detectives.
The Touch Detective series shows Mackenzie as this, but still in the Kid Detective phase.

Western Animation

Inspector Gadget's niece Penny would likely have grown up to be one of these, after all the practice she had "helping" her uncle solve his cases. Subverted, in that Gadget was chiefly useless and Penny did most of the work.