The four writers (and presumably potential showrunners) are Max Borenstein, Brian Helgeland, Jane Goldman and Carly Wray.
Borenstein is heavily involved in Legendary's new "Monsterverse", writing the scripts for Godzilla and Kong: Skull Island. He is feted in Hollywood writing circles for Jimi, an unproduced biopic about Jimi Hendrix. He is currently working on the Godzilla sequel, King of the Monsters.
Helgeland is an Academy Award-winner, noted for his work on L.A. Confidential and Mystic River. Intriguingly, his scripts include the Heath Ledger 2001 medieval musical A Knight's Tale, which bears uncanny similarities with George R.R. Martin's Song of Ice and Fire prequel The Hedge Knight (1998).
Jane Goldman is an experienced SFF screen writer, having written the scripts for Stardust, Kick-Ass, X-Men: First Class, The Woman in Black, Miss Peregrine's Home for Peculiar Children and both Kingsman movies. Goldman has a high hit rate and is the only one of the four writers to be based in the UK, presumably where the spin-offs will continue to be filmed (if they require the same scenery).
Carly Wray has worked as a writer and producer on Mad Men, Constantine, The Bastard Executioner, The Leftovers, Mindhunter and Westworld. Her work on The Leftovers and Westworld gives her the most experience of having previously worked with HBO.
Borenstein and Helgeland are developing ideas independently, whilst Goldman and Wray are developing their scripts with input from George R.R. Martin. Game of Thrones producers David Benioff and D.B. Weiss will be attached as producers and advisors, but will not be directly involved since they are starting work on the eighth and final season of Thrones for next year and will then be developing a feature film adaptation of the novel Dirty White Boys for Fox.
HBO are apparently not set in any final decisions about the spin-off project: they may like all four ideas and commission all of them, or dislike all of them and go back to the drawing board. It's worth noting at this point that George R.R. Martin retains the rights to any other material set in Westeros, so the writers will have to also win his approval as well as HBO's to move forwards.
The most likely spin-off project is either a continuing series or series of films based on Martin's Dunk and Egg novellas (starting with The Hedge Knight). There is already material available to be adapted and the stories have a (somewhat) lighter, more self-contained narrative that will make them, in theory, easier to sell to audiences. However, only three of a potential twelve stories in the series have been written. It could be, with the main series occupying his time for the next few years (at least), Martin has found the TV option as an alternate way of getting these stories out through other writers.
HBO would likely be most interested in a prequel series based on Robert's Rebellion, featuring younger actors playing key Game of Thrones roles. Indeed, they already have Robert Aramayo and Aisling Franciosi cast as Eddard and Lyanna Stark from the main series (although a big-budget, ongoing series may feature some recasting). However, Martin has always been somewhat opposed to telling this story in detail, believing it would only be "filling in the blanks" as the key parts of the story have already been revealed in A Song of Ice and Fire and Game of Thrones themselves. HBO may attempt to persuade Martin of the merits of the project by
More daunting is the idea of doing a story based on the Dance of Dragons, using Martin's short stories The Rogue Prince and The Princess and the Queen, as well as the history outlined in The World of Ice and Fire (not to mention a much more detailed account of the conflict, which Martin has written for a future spin-off project called Fire and Blood). This would probably need to be a feature film or a trilogy, and with its focus on massive dragons and aerial combat, might be a good fit for Borenstein (whose upcoming workload is also quite full, making it less likely he would commit to a TV show for several years but could work on a movie project).
It's very early in the process, but it's quite likely that HBO would like something ready to air in late 2019 or early 2020, after Game of Thrones' final episode airs somewhere in the late summer of 2018.