I don't know how unique or different this would be from Enterprise, but I've always wanted to see what it was (or would be) like on the very first Federation starship constructed after the birth of the Federation and the end of the Romulan War.
The Federation and Starfleet are in new territory: no longer embroiled in war, they are now a new entity dedicated to exploration. The first ship designed and crewed by members of all the founding worlds would be a historical endeavor and would be a great setting for conflict and character development.
Obviously, there wouldn't be Vulcans on this ship (unless Spock's status as the first Vulcan in Starfleet is ignored or the series is set in the Abramsverse) but there would be Terrans, Andorians, Tellarites, and maybe a few other races. If memory serves, the Alpha Centaurians were founding members. They'd be an interesting concept to explore: originally human, they've evolved due colony life.
These races' governments may have become the Federation, but how would individuals feel about working together? Surely there would be distrust and drama.
As for the ship itself, I'd go with a play on that NX refit from Doug Drexler and call it (and the show) Phoenix, named after Cochrane's ship.