There two ways to go with a character, with what's uniquely Eberron, though in my opinion one is still borderline "vanilla" D&D.
The most unique race in Eberron is the Warforged, a magically created machine-man of sorts. There have been Golems or Constructs in other games, but they were fairly unthinking. This race is the embodiment of Eberron and its magic-saturated world.
The class that is new to Eberron and very unique is the Artificer, which can create magic items or cast spells by understanding their underlying energies. They're not a wizard, they're not a cleric, but they're somewhere in between.
These two combined is a class/race combo unlike anything in any other D&D game of the past and what I would consider iconic for the setting.
The other unusual aspect of Eberron the greater emphasis on psionics, instead of just divine or arcane magic. This has been done in earlier game settings, so it isn't that new. There are races (e.g., Kalashtar) which really work well with Psionics and are new.
Ultimately, the Eberron setting is about magic being so ubiquitous that it's in everything and isn't as fantastic as other worlds. Additionally, it's very pulp-noir, encouraging a world more like Indiana Jones' where you're finding ancient cultures and artifacts. My own label for the world though is "mana-punk" which is where Steampunk meets Magic.
In any case, this is my favorite campaign world yet created, and just embracing the shift to include technology and adventure makes any combination of race and class quite fun to play.