The doctor (Tennant's Seasons I believe) did have a Time Lord daughter named Jenny. She was created by them taking a sample of his blood at the beginning of the episode and them mixing his DNA with a randomly selected individual's DNA which had previously been stored in the machine. She grew into a woman and stepped out of the machine and was proven to be able to regen in the same episode. You could do a totally different spin off by playing her and having her either find a Tardis all her own or somehow find a way to build one? There were several broken down ones throughout the series.