Amy and Rory: Pond Life
Mother and Father of River Song, Amy and Rory were sent to Earth for a period of time by the 11th Doctor in hopes they might live. This was between Doctor Who Series 6 and 7, and covered an important gap in time that saw Amy become a model and eventually led to a falling out between her and Rory. A short from the BBC titled "Pond Life" saw them befriending an Ood, frequent phone calls from their son in law and occasional visits from River. Spin-off material, we think.
Susan Foreman: I.M Foreman
Susan Foreman traveled with the 1st Doctor and swept her school teachers Ian and Barbara Chesterton into their adventures. She's the Doctor's granddaughter from his first marriage on Gallifrey, and stole a TARDIS with him to explore the universe. After she fell in love with a human, the Doctor left her behind to find her own life. We can picture new companions of the Doctor going to her for a cuppa tea and advice. Sarah Jane definitely knows of her on The Sarah Jane Adventures.
Romana (Fred): Journeys of E Space
The noblest Romana of them all, so much she traveled with the Doctor twice! Some fans even suspected a romantic relationship between her and the Doctor; Lalla Ward and Tom Baker were married on set for a while. After Mary Tamm regenerated into Lalla Ward, Romana was sent to E Space, a dimension outside our own with a green tint, with K9 by choice. Unaired adventures between her and the Doctor such as "Shada" or her journey to become president of Gallifrey in the Big Finish audios would be interesting to see.
Jenny (The Doctor's Daughter): An Awful Lot of Running
Jenny is the Doctor's daughter, born from a stolen tissue sample on the planet Messalaine. She's engineered to be the perfect soldier and was thought to have died on the Doctor Who episode "The Doctor's Daughter." However, we see her stealing a ship and flying away after the Doctor leaves. She would explore the universe as she was born yesterday, and get to know herself as a part Time Lady! Big Finish audios have a series for her called Jenny starring Georgia Moffet, and it's well worth a listen.
Harriet Jones (Yes, We Know who she is): Battle for Minister
Harriet Jones battled Slitheen with the 9th Doctor on the Doctor Who episodes "Aliens of London/The End of the World" and protected Rose Tyler, gaining his admiration. She runs for Prime Minister and leads Britain into its golden age. When she destroys the Sycorax on the Doctor Who episode "The Christmas Invasion" because she wasn't able to stop her men from being killed, the Doctor ends her career with the simple whisper of 'don't you think she looks tired?' Harold Saxon (the Master) takes over from her, but the spin-off would show the politics of the take over and if it was willingly...or through possession.
Brigadier Lethbridge Stewart and Kate Stewart: UNIT HQ
Brigadier Lethbridge Stewart was one of the first companions of Classic Doctor Who. He went on to have a daughter called Kate Stewart and she took over UNIT after him. The show would be her growing up and learning the ways of alien-fighting, with lots of stories about the Doctor from her father. She'd later create the Black Archives, an alien weapon storage of which the Doctor doesn't approve. Kate Stewart has appeared in many Big Finish audios about UNIT, which show the alien-fighting troops dealing with the Master.