film prototype: Tarzan Finds a Son! (film, 1939) 

  After several Johnny Weismuller/Maureen O'Sullivan films for MGM 
  beginning in 1932 (and a number of other Tarzan films at other studios 
  beginning in 1918), MGM decided that the thing to do was add a child to 
  the mix in the person of Boy (Johnny Sheffield), even though parents Tarzan 
  and Jane technically were "living in sin"!. As noted in the title,
  Boy is "found" in the jungle, having survived a plane crash -- 
  not born to Jane, a storyline that probably would have been censored in
  late-1930s Hollywood.

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The Adventures of Ozzie and Harriet (ABC,1952-1966) 
  Stanley Livingston (1958-1959), Barry Livingston (1961-1962), recurring
  guests (usually referred to with their own first names, Stanley and Barry).
  Some shows had other 'guest kids', and many with Stanley and/or Barry had
  additional kids. Shows with these extra guest actors were very common in
  the last four or five of the show's 14 seasons.

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All in the Family (CBS, 1971-1979) and Archie Bunker's Place (CBS, 1980-1982)
  Stephanie Mills (Danielle Brisebois, 1978-82)is a niece abandoned by
  father, adopted by the Bunkers.

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The Benny Hill Show (BBC, ITV, Thames Television, 1955-1989) 
  In last two seasons, 1988-1989, the show added several small children as 
  "Hill's Little Angels".

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The Brady Bunch [The Brady Brood](CBS, 1969-1974) 
  Cousin Oliver (Robbie Rist [Robert Anthony Rist], 1974)is a Brady cousin who 
  comes to live with the Bunch. 

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Bonanza (NBC, 1959-1973)
  Jamie Hunter (Mitch Vogel, 1970-1973) is an orphan taken in by the Cartwright 
  men and later adopted (changing his character name to Jamie Cartwright).  
  Gary Brumburgh (gr-home@pacbell.net) notes in his biography of Mitch Vogel 
  posted on IMDb that a young teen may have been needed in the storylines, as 
  "Michael Landon, at age 34, was a bit long in the tooth to be receiving 
  fatherly guidance by patriarch Ben." 

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Chico and the Man (NBC, 1974-1978)
  Raul Garcia (12)(Gabriel Melgar, 1977-78), a stowaway in trunk when Ed Brown
  (Jack Albertson) and Louie (Scatman Crothers) return from Mexico, is later 
  adopted by Ed. 

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Diff'rent Strokes (NBC, 1978-1985; ABC, 1985-1986)
  Sam McKinney (9)(Danny Cooksey, 1984-86) is the son of Maggie (Dixie Carter, 
  then Mary Ann Mobley) whom Mr Drummond marries in February 1984.  

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Eight is Enough (ABC, 1977-1981)
  The first episode of Season 5, the final season, was And Baby Makes Nine
  (29 October 1980), in which Susan Bradford [Stockwell] (Susan Richardson)
  introduces the title-breaker baby.  Apparently, Eight Isn't Enough after
  four seasons! 

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Family Matters (ABC, 1989-1998) 
  In the Trivia section for Jaleel White (Steve Urkel), IMDb documents the fact that
  Urkel's character "was supposed to be a guest-star, but proved to be so popular that
  he stayed on as a regular." It is said that, after filming his original scene,
  "50 frat boys in the audience started chanting, 'Urkel! Urkel! Urkel!'."
  According to IMDb, Urkel's first appearance was in S1Ep4, "Rachel's First Date"
  (13 October 1989), then he reappeared in S1Ep7, and was more or less a regular from
  then on. (One viewer comment on IMDb notes that most people probably don't even know
  the official series name, as it came to be known popularly as 'The Steve Urkel Show'.
  [IMHO, this show was so derivative and un-original in its stories and scripts that
  it could not have survived without the flamboyant, breakout originality of Steve Urkel.]

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Family Ties (NBC, 1982-1989)
  Andrew Keaton (Brian Bonsall, 1986-89) arrives as a new baby in the original 
  Keaton family.

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The Golden Palace (CBS, 1992-1993) 
  Oliver Webb (13)(Billy L. Sullivan, 1993) is an example of a reverse of 
  this "formula": this one-season spinoff of The Golden Girls
  started with Oliver Webb, whose mother had been entered into rehab (father 
  in jail) and Roland Wilson (Don Cheadle) had become his temporary foster 
  parent; mother comes and takes Oliver back in mid-season; Oliver is not seen 
  again.

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Good Times (CBS, 1974-1979) 
  Penny Woods (Janet Jackson, 1977-1979), a victim of child abuse, becomes 
  adopted daughter to Willona (Ja'net DuBois).

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Growing Pains (ABC, 1985-1992)  
  Luke Brower (Leonardo di Caprio [Leonardo Wilhelm di Caprio]) is a homeless
  boy, student of Mike Seaver (Kirk Cameron) who is beginning his first job as
  a teacher in an inner-city school in Season 7 (1991-92); Luke moves in with
  the Seaver family. Luke is really the second "child add-on" to the
  series, as Maggie gave birth to a girl at the beginning of Season 4
  (Christine Ellen 'Chrissy' Seaver, played as an infant and toddler by Kelsey
  and Kirsten Dohring and later as a 5-year-old by Ashley Johnson). 

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Happy Days (ABC, 1974-1984) 
  Chachi/Charles Arcola (Scott Baio [Scott Vincent James Baio], 1977-84),
  cousin of the Fonz (Henry Winkler) comes to live with him, ends up marrying
  Joanie Cunningham (Erin Moran) and continuing into a (TV) spinoff.  

    "Passages", 8 May 1984
  The Fonz (Henry Winkler) considers adopting his Big Brother program 
  "little brother" Danny (Danny Ponce) in one of the last-season 
  episodes (hoping for a sequel/spinoff?). 

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I Love Lucy (CBS, 1951-1958) and The Lucy-Desi Comedy Hour (CBS, 1958-1960) 
  Perhaps we can't really count this as intentional, but it did boost the 
  ratings and may have been a prototype for TV: after 11 years of marriage, Lucy 
  (Lucille Ball) and Ricky (Desi Arnaz) have a "blessed event"
  (propriety at the time forbade using the word "pregnant")
  in "Lucy Goes to the Hospital" (19 January 1953, the same day that
  Lucille Ball, in real life, gave birth by Caesarian Section to their son
  Desi Arnaz Jr). The boy is named Little Ricky/Ricky Ricardo Jr (played by Keith
  Thibodeaux [Richard Keith] as a toddler and youngster, 1956-60). 

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Malcolm in the Middle (Fox, 2000-2006) 
  New baby Jamie joins Malcolm's family in the birthing episode, "Baby: Part 2",
  18 May 2003 and is seen in most of the following episodes (at first uncredited,
  then played by toddler twins James and Lukas Rodriguez [2004-2006], sometimes 
  with screen credit, sometimes not).

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Mork and Mindy (ABC, 1978-1982)
  In a twist on the usual addition of an actual child actor, Jonathan 
  Winters (!) is added in 1981-1982 season as Mork and Mindy's son 
  "Mearth from Earth"; like his father Mork, Mearth ages backwards,
  so he arrives in adult form.

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My Three Sons (ABC, 1960-1965; CBS, 1965-1972)
  Robbie Douglas Jr (Danny Todd [Daniel Todd], 1970-1972), Steve Douglas II 
  (Joseph Todd, 1970-1972), Charley Douglas (Michael Todd, 1970-1972) play
  triplets born to original son Robbie and his wife Katie. The show lasted so 
  long that original cast members became adults and spawned a new generation.
  Todd brothers are actual triplets themselves. 

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Two and a Half Men (CBS, 2003-2015) 
  This is one of the more interesting and complex "attempted saves" 
  in television history. The show started its unusually long 12-year run 
  already featuring an adorable child (Angus T. Jones as Jake Harper, age 10)
  in a title role. When Jones grew up and acquired religious attitude, however
  (this is well-documented on the Web, and does not need to be revisited in
  detail here), the producers replaced him in Season 9 with a long-lost
  niece of the late Charlie Harper (Charlie Sheen), another exit from the
  series due to personality implosion (again, readers are advised to read more
  on the Web). She remained through Season 12, when it finally was decided
  to "restore" the 1/2 man of the title in the person of 
  Louis (Edan Alexander), an adopted 6-year-old son for Walden Schmidt
  (Ashton Kutcher) and Alan Harper (Jon Cryer) who - wait for it - decided to 
  get married (yes, to each other!) in order to elevate their chances to 
  adopt, as couples are given preference to single-parent adoptions.
  This move provided new story material about the adopted son, the gay
  marriage, a "marriage of convenience" (so that they could
  adopt), straight men in a gay marriage "cheating" with
  beautiful women, and so on. (Despite all these new angles, this was
  the show's last season, and was a truncated one at that - 15 episodes,
  as compared with the more typical 24 of previous seasons.)

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What's Happening! (ABC, 1976-1979)
  Little Earl Babcock (9)(David Hollander, 1978-1979) arrives as a new neighbor 
  who becomes wildly attracted to Raj's bratty little sister Dee (Danielle Spencer).


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Who's the Boss (ABC, 1984-1992)

    "The Kid", 23 October 1990
  William 'Billy' Napoli (Jonathan Halyalkar) is added as a family 
  friend who lived with his grandmother, and came to live with Tony Micelli
  (Tony Danza) and Angela Bower (Judith Light) when his grandmother was 
  recovering from hip surgery. He was there for most of the episodes in the 
  next-to-last season, then he was gone for the last season. In 
  "Seer of Love" (28 September 1991, first show of final 
  season [8]), it was explained that his grandmother had completely recovered
  from her surgery and he could go back to live with her again.  (Some 
  bloggers on IMDb have noted that he was a cute kid, but they didn't like him 
  very much.)