In "The Invasion", Professor Watkins and his daughter Isobel are Suspiciously Similar Substitutes written in when the actors of Professor Travers and hisdaughter Anne turned out to be unavailable for filming. Professor Travers and Anne are said to have moved to America, and Professor Watkins and Isobel fill their shoes extremely well. This is quite possibly lampshaded by the fact that the Doctor and companions mention repeatedly that they were hoping to meet Travers and Anne before finding out that the pair are gone, even though they are otherwise unimportant to the story.
Colonel (later Brigadier, later still Sir) Lethbridge-Stewart resembles Colonel Breen from Quatermass and the Pit. So much so that originally Julian Glover from the then-recent movie adaptation would have played him had the original actor not dropped out.
The last story of the sixth season featured the War Chief. He's an evil, megalomaniac Time Lord who dresses in a dark Nehru jacket, sports Facial Hair of Evil and knows the Doctor from their days on Gallifrey. He's extremely camp, has no concept of personal space and offers the Doctor a half-share in the universe. Fast forward to Season 8, where we are introduced to a new regular villain who's an evil, megalomaniac Time Lord, dresses in a dark Nehru jacket, wears a Beard of Evil and— oh, you get the point. Fanon has even identified them as the same character.
Ernst Stavro Blofeld has been an enduring influence on the Master; what with the Nehru jackets, the kitty-stroking in "Survival", the elaborate deaths for unwary henchmen, etc.
Jack Harkness has a number of similarities to one of the Eighth Doctor's Doctor Who Magazine comic-strip companions, Fey Truscott-Sade, who like him is a morally-ambiguous secret agent with Extreme Omnisexual tendencies, who flirts with the Doctor a lot. The only major difference is that Fey is female, although very gender-ambiguous.
Companions Amy Pond and Rory Williams share quite more than a few traits with Sally Sparrow and Larry Nightingale from "Blink", a story Steven Moffat wrote before taking over as the head writer.
Lumpy the brain-damaged "good" Time War Dalek in the BBC online game The Doctor and the Dalek is a clear expy of Rusty the brain-damaged "good" Dalek from "Into the Dalek".
"The Return of Doctor Mysterio": The Ghost is clearly based on Superman with many of the same powers, Clark Kenting using glasses, and pining for a reporter, who has no idea the bumbling ordinary guy is the masked superhero. Even the origin of his powers is similar. Supes gets his abilities from a yellow sun, while Grant has accidentally swallowed a wish-fulfilling gem made inside a star. And he loved Superman comics as a kid, so the powers he manifested make sense.
The Twelfth Doctor's companions in Series 10. Bill Potts, a working-class London girl associated with chips, is similar to the Ninth/Tenth Doctor companion Rose (her name even has some similarities to the name of Rose's actress, Billie Piper). Nardole, a Plucky Comic Relief, secretly hyper-competentServile Snarker who is the Doctor's best friend despite being the more minor character, fulfils similar function to the iconic Fourth/Tenth Doctor companion K-9. This was all part of Revisiting the Roots.
Graham O'Brien is a soft-spoken working class man in late middle age who has his perspective expanded by travelling with the Doctor. These traits are all shared with Rory's father Brian, also created by Chris Chibnall.
