Or possibly the Blue Peter monster design competition.
Getting an adventure from an outside source - a DWAITAS supplement, another Who source like the books, comics or audio plays, a different game, some random geek's rambling Who blog - would fit here.
But one particular option is to canvass the players and see what adventure ideas they come up with. They probably have ideas for what they'd like their characters to see and do (which may or may not resemble what the characters themselves would want...) and might have suggestions for monsters and the like too.
The ultimate step is to let someone else take a turn GMing an adventure. Normally this is one of the players as they know the series, which can prove a fascinating insight into how the game works.
8: I was tempted to get someone else to write this article...
Taking over the player's normal character could be useful, especially if the player controls a key character like the Doctor, but also means having to act like them, which can distract from the game.
Alternatively, park their character somewhere (visiting family, recovering from a bump on the head, abducted by aliens, lost in a freak temporal vortex) or they'll have to trail along with the group as an NPC. (One possibly complicated option would be to separate them for a subplot, which you run.)
You, meanwhile, create a suitable PC for a single adventure, probably based on the episode's setting - someone like Gwyneth, Harriet Jones (MP Flydale North), Donna in her first appearance, Tallulah, Joan, Astrid Peth or Midshipman Frame, Jackson Lake or Rosita, Liz Ten or even Shakespeare or Vincent. They might go on to be recurring characters, or might die heroically - this is Doctor Who after all.
Or, bring in a PC-level NPC that the regular cast can get along with. Like a Sarah Jane or Captain Jack guest appearance, or the times the Brigadier really gets stuck in rather than showing up with the cavalry. Teaming up with a villain could be fun too, if the players don't mind.
Be ready to answer rules and setting questions, brainstorm ideas and so on. Equally, be ready not to.
Discuss possible long-term ramifications of the plot as well, if any - the guest GM could possibly run a completely contained adventure. The first time I ever handed the reins to a player, he checked in advance if a friendly NPC having a major freakout would work. And if the game goes off in a direction you'd rather not feature in later episodes, you can always say it was a spinoff story rather than something in the series itself...
Example: The War That Never Was
by Emily's Player
In a ruined paradoxical present-day Earth, the group are separated when alien invaders in heavy space armour capture Emily, and the Doctor and Richard work with a survivor to free her and find the source of the paradox allowing this impossible invasion to happen.