Supposedly, Christopher Eccleston wished Steven Moffat well with his plans for the 50th anniversary special for 'Doctor Who', but he had no interest in reprising his role as the first Doctor to be seen in the new series in 2005. When he first appeared on the scene, he had just recently regenerated and only first saw what he now looked like in the hall mirror at Jackie Tyler's flat. So I imagine what Moffat had in mind was to have Eccleston around just long enough to act out the regeneration from John Hurt's War Doctor. (Perhaps even one scene to show the connection to the Nestene Consciousness? Although I think that it was the TARDIS - as usual - who put him on the track of that by landing in London of 2005.)
As it plays out in the special, we see the War Doctor begin his regeneration cycle, with a slight change to someone who looks similar to Christopher Eccleston (if not some old footage of that actor) but the scene cuts away before the cycle is completed.
Still we saw enough to make two O'Bservations.
One: This regeneration was similar to that of the First Doctor into the Second in that his body finally knew it was time to renew. After so many countless years in battle, the culmination of the Time War was a signal that his services as the War Doctor were no longer needed. He had become stretched and thin, as a certain Hobbit would put it; and while he was never going to find rest, at least he would be renewed and refreshed for whatever new problems the Universe held for him once he regenerated.
(The actual regeneration process of fire and light remained similar to that which chronologically began with the Eighth Doctor.)
Two: So where did the DNA template for Eccleston's Doctor come from?
Because his entire career was focused on the Time War, it's possible that the War Doctor came into contact with another Time Lord (or even just a normal Gallifreyan - there is a difference!) who bore similar features.
But there's a whole universe of other options out there....
"This is the truth, Doctor:
You take ordinary people and you fashion them into weapons...,"
"The Journey Home"
Davros more than likely crossed paths several times with the War Doctor during the Time War and saw for himself that this was the case with the companions that accompanied him during the conflict. The War Doctor may have even chosen them for the skills they already possessed.
Some spoilers ahead......
One candidate might be Frank Carter, a young hot-head social activist who was suspected in the murder of the dentist Dr. Morley (as seen in the 'Poirot' mystery "One Two Buckle My Shoe".) Before he ran afoul of Inspector Japp, it could be that Carter spent some time travelling with the War Doctor. His experiences may have soured him on Life after seeing the worst the Universe had to offer during the Time War.
A better candidate would be a fellow named Claude, a mutant seen in several episodes of 'Heroes'. Although 'Heroes' veered off into a parallel dimension with the arrival of Future Hiro, it began in Earth Prime-Time. So all of the characters would have already been established in the main Toobworld even if they didn't appear until after the divergence.
Claude would have made for an excellent ally for the War Doctor considering his mutant ability was the power of invisibility. But eventually his shady character would have caused a rift in the relationship he had with the War Doctor and they would have parted ways. Of course, he may also have died during the Time War - I don't think the survival rate for any of his Companions was very high during those dark times.
And then there's one last candidate, one who was more likely to be an atagonist rather than a Companion during the Time War - Malekith the Accursed.
Malekith the Accursed is the king of the Dark Elves of Svartalfheim, as seen in "Thor 2: The Dark World" and in the Marvel comic books. Although he's primarily in the Cineverse as a live-action character, and never seen in TV, Malekith belongs to the alternate TV dimension of Marvel Toobworld. Still, Thor is part of Earth Prime-Time, thanks to appearances in 'The Incredible Hulk' and 'Hercules: The Legendary Journeys'.
Based on those two portrayals, we know that Thor has the ability to alter his physical appearance. So it's pozz'ble, just pozz'ble, that Thor now looks like Chris Hemsworth in the main Toobworld as well. If so, his adventures in the movies (three so far) could also have happened in Earth Prime-Time, with adjustments so that they didn't cause any major discrepancies. (Like that battle in New York City in the "Avengers" movie is right out!)
If the War Doctor came into contact with Malekith during the Time War, there are several factors that may have proven desirable to his body's imperative to collect the DNA. Physical appearance would be the least of it, but even so the adaptation to a more human-looking version couldn't get the ears right.
Also, there was Malekith's strength in character and purpose and determination which added up to being factors in his choice. But like the choice of Nasca by the Second Doctor's biology, Malekith's ruthless nature would have been discarded. Otherwise Eccleston's Doctor would have had no qualms about destroying all of Mankind along with the Daleks in "Parting Of The Ways". (Memories of the true solution to the Time War may have been at play.)
I like to thnk that if Malekith was involved in the Time War, he became the Could Have Been King.......
Perhaps there are other options out there among the characters played by Christopher Eccleston. But these are my three favorite choices......