My thoughts:
Mark Gatiss
Long-term DOCTOR WHO megafan, starting his association with the programme by writing the (pretty good) novel Nightshade in 1992. Known for his work on the TV series THE LEAGUE OF GENTLEMEN, on which he was a writer, actor and co-creator. Has recently co-created and co-showruns SHERLOCK alongside Steven Moffat. He has written six episodes of DOCTOR WHO since 2005, which have had a 'mixed' reception. However, I'd argue his last two stories were actually pretty good.
He has DOCTOR WHO experience and, crucially from a BBC perspective, experience as a showrunner. He is perhaps the most logical choice to take over WHO when Moffat leaves. What is a major strike against him is that he has just signed up to play the recurring role on Tycho Nestoris on HBO's GAME OF THRONES. Whilst not a massive role, it does require him to appear in multiple episodes across Seasons 4-6 (at least), some already filmed. This may preclude him from taking over DOCTOR WHO if Moffat leaves in the next 1-2 years. However, as a small role it could be recast if necessary.
Toby Whithouse
An experienced showrunner, producer and writer, best-known for creating and producing BEING HUMAN for the BBC (which has spawned a very successful American remake). He has also created and run the shows NO ANGELS for Channel 4 and the upcoming THE GAME for the BBC. He has DOCTOR WHO writing experience and seems to be well-regarded by the fans. Another logical choice.
Gareth Roberts
Another experienced writer and DOCTOR WHO fan, cutting his teeth with the New Adventures novel The Highest Science (which was nearly remade for the modern series until being replaced by Planet of the Dead) and contributing multiple scripts and novels in the franchise. He doesn't have showrunning experience per se, though he played a major role in the writing of the first season of THE SARAH JANE ADVENTURES. He has also been a script editor for multiple soap operas, including EMMERDALE, BROOKSIDE and CORONATION STREET. Would probably be a good choice, but the BBC might prefer someone with more production experience.
Neil Cross
Another experienced showrunner, having created the massive international hit LUTHER starring Idris Elba. He has strong international writing experience and is a collaborator of Guillermo Del Toro, having worked on his aborted At the Mountains of Madness project and done rewrites on Pacific Rim before writing Del Toro's next script, Midnight Delivery. He also wrote the well-received horror movie Mama, as well as several novels (one of which was longlisted for the Booker Prize).
Cross might be a stretch: he lives in New Zealand and prefers to travel in bursts to work on projects. He may not want to physically move to the UK for several years to work as a WHO showrunner full-time. He also has a very good and booming Hollywood scripting career, which likely pays a hell of a lot more.
Still, under Cross's stewardship we could very well get a Guillermo Del Toro-directed episode starring Idris Elba (maybe as the next Doctor?), which I'd be quite happy with.
Howard Overman
The creator and showrunner of the very well-received MISFITS, and the co-creator and co-executive producer on ATLANTIS. He has showrunner experience but hasn't worked on DOCTOR WHO before. He's the most logical choice if the BBC decide they want some fresh blood.
As for a mental, will-never-happen-but-hey-what-the-hell choice:
J. Michael Straczynski
The creator and showrunner of BABYLON 5, the showrunner of JEREMIAH and the scriptwriter of numerous films (including the very good Changeling, starring Angelina Jolie), animated series (HE-MAN! GHOSTBUSTERS!) and comics (everything from a well-received run on SPIDER-MAN to the recent BEFORE WATCHMEN project). Notably, a big fan of classic WHO, which he cited (alongside THE PRISONER and BLAKE's 7) as a major influence on BABYLON 5's story arc structure.
It won't happen, of course. He's American and lives in LA, though has said over the years he wants to retire to the UK. He hasn't written for WHO before either and hasn't indicated he wants to. Still, he'd also be a good choice because of his writing speed (he can write a 44-minute TV episode in two days) and also his production management skills which helped prevent both his previous TV shows from going over-budget or over time, ever. More importantly, he has just committed to producing (and apparently show-running) SENSE8, a new Netflix TV series created by the Wachowskis.
(note: Joss Whedon, though also a DOCTOR WHO fan, is under exclusive contract to Disney for the next four years or so, so is out of even the most optimistic running)