This morning, William Hill currently place Ben Whishaw as favourite, at 6/1: the actor is probably best known worldwide as the character of Q in the Daniel Craig era of James Bond films, or for the younger audience as the voice of Paddington Bear in the 2014 film (and this year's sequel), and on recent television in London Spy . Next up is Richard Ayaode at 8/1, the quirky comic actor/director is best known in the UK for The IT Crowd, his travels around the world in Travel Man, and will be the host of the revived series of The Crystal Maze this year. Also at 8/1 is another name long-associated with Who-odds, Rory Kinnear (another modern James Bond stalwart as Bill Tanner), and Rupert Grint (or as most people know him, Ron Weasley in the Harry Potter films!). The first actress on their list is Broadchurch's Olivia Colman at 20/1 - though ITV's Good Morning Britain reported that Miranda Hart was in fourth place at 8/1. The current list can be found via William Hill's special bet "To Replace Peter Capaldi as the next Doctor Who?".
Both Betfair and Paddy Power offer up Sherlock's Moriarty, Andrew Scott, as their joint favourite, alongside Ben Daniels (Law and Order UK, House of Cards, Star Wars: Rogue One), and Lara Pulver (Spooks, Sherlock).
During an item on the departure of Capaldi on BBC's Breakfast this morning, viewers' suggestions were read out, including names such as Hayley Attwell or Eddie Redmayne, Colin Salmon, David Harewood, Helena Bonham Carter, Gillian Anderson, and Danny Dyer.
Anticipation will of course continue until the eventual new Doctor is announced by the BBC - how and when this might be is unknown at present (Capaldi's announcement in 2013 took place on a special live television programme!), but may be held off until the last series starring Capaldi has been broadcast - it will be on air between 15th April and 1st July (barring pre-emptions such as for the Eurovision Song Contest), with filming for the next series under the helm of new lead writer Chris Chibnall expected to commence not long afterwards.
Update 1st Feb: the Radio Times has reported that filming for Series 11 would not commence until Spring 2018, so an announcement of a new Doctor might not occur until next year!