The Doctor's loyal tin dog K9 (John Leeson) has a numerical link to a monster from Tom Baker's Doctor Who era. Introduced in the 1977 serial The Invisible Enemy, K9 became a regular fixture of the Tom Baker era, until he was written out in 1981's Warrior's Gate. The Doctor has regularly given out K9s as gifts to his companions, as Leela (Louise Jameson), Romana (Lalla Ward), and Sarah Jane Smith (Elisabeth Sladen) have all found themselves with a K-9 of their own.
Although it's a pun, K9's name does have a strange numerical link back to the villain from the Fourth Doctor's very first adventure, Robot. In that story, the newly regenerated Doctor helped Sarah Jane Smith and UNIT unearth a conspiracy involving an elitist scientific group and an experimental robot called K1. It was a Doctor Who twist on King Kong that ended with the giant robot falling in love with Sarah and taking her hostage in his giant claw. The similar naming structure suggests that K9 could be a descendant of K1 via the two scientists that created them, Professors Kettlewell and Marius.
How Doctor Who's K9 Could Be Connected To K1
K9 was created by Professor Marius as a replacement for his own dog, who was unable to travel with him to the Bi-Al Foundation, a medical facility housed on an asteroid. Originating on Earth, it's possible that Marius could have some family connection to Professor Kettlewell, who built K1 centuries earlier. Constructed from living metal, K1 had a strong bond with Kettlewell, who he essentially saw as his father. The tenderness between K1 and Sarah Jane was very similar to how Marius, and the Doctor's companions all treated K9, even if the robot dog wasn't as emotionally forthcoming.
Kettlwell was killed during the events of Robot, so he wouldn't have been able to continue his research. However, it wouldn't be surprising if someone did decide to continue his research, eventually leading to K9 being the ninth model of Kettlewell's original work. As the Doctor meets K9 for the first time in the year 5000, it suggests such an evolution of Kettlewell's work took a very long time. However, given that K1 almost caused the nuclear annihilation of the entire human race, perhaps Doctor Who's UNIT and the world's governments decided to suppress information about K1 for a very long time.
What K9 Is Really Named After
K9 is obviously a sci-fi pun on the part of the robot dog's creators Bob Baker and Dave Martin. However, as Marius built the robot to replace his own dog, Marius' own pun could connect back to the legacy of Professor Kettlewell and K1. Contrary to popular belief, K9 is only really a dog in appearance, he's essentially a super-intelligent artificial intelligence housed inside a mechanical dog. Therefore, it could easily be explained in-universe that K9 is an extension of Kettlewell's work with A.I. that's been given a canine twist.
Although K9 was a hit with kids, 1980s producer John Nathan-Turner felt that the super-intelligent robot, combined with the Fourth Doctor and Romana, added to a smug intellectual tone that was off-putting to viewers. This was why, in 1981, K9 was finally written out of the show and given his own spinoff pilot, the unsuccessful K9 & Company. Since K9 returned alongside David Tennant's Tenth Doctor, Doctor Who's robot dog continues to be linked to the legacy of the show, and may even be continuing the legacy of Professor Kettlewell's tragic robot K1.