A simpler alternative might be to grant all PCs in the campaign a bonus feat at 1st level. I'm not sure how much that would mess with the balance of the game.
I would like to see the Dragonmarks being less class-specific in their utility than they were in 4e.
I think the bonus 1st level feat is the way to go. 3E Eberron had action points, so there's a history of a unique perk for Eberron games. As far as what to do if the group isn't playing with feats... handle it as per standard: Don't say it's a feat, state that all PCs get an additional +2 stat advancement at first level that can be swapped for a lesser Dragonmark or any other equivalent. Include the Dragonmark progression as feats, but state that those specific feats aren't optional in Eberron, but the others are. Done.
If you don't like the free-wheeling stat advancement at 1st level, then port-forward action points. A 1st level PC gets either a Dragonmark or an action point pool. The pool contains one point (feats or level advancement could optionally modify) that is replenished every long rest. Action points work pretty much like Inspiration, except they aren't under the DM's control. There's probably some balance to be done there: refill per short rest, 2-3 points per long, etc. but the idea is simple enough. If you don't mind feats, the make Heroic Actor a feat that grants the above.
With skills and tools being much less class-specific in 5e than previous editions, I actually think that the 4e "I'm just better at stuff" mechanic (which I disliked for 4e) finally shows some real promise. You could even do stuff like Lesser Dragonmark of Healing "Treat Cure Wounds as though it appears on your class spell list. If Cure Wounds already appears on your class list or you cannot cast spells, you gain the ability to cast Cure Wounds once per long rest without using a spell slot. Cast this way, Cure Wounds is considered as the highest level slot available to a multi-class caster of your level." I'd prefer more "double your proficiency bonus" type things, when possible, but that would certainly work (and feels balanced).
I
don't like the idea of using backgrounds. Doing that means that you can't be a Cannith scholar (or criminal, for that matter). It's no longer a background -- it's a subrace. There could be house-specific backgrounds, though.
Actually, I don't really like any option that locks you into the Dragonmarked or not at first level. Most of the Marked IMC have been late-bloomers. Even if that's not the standard, it should be handled by the mechanics. That pretty much leaves feats or "prestige" classes, which I'm not real excited to introduce into 5e.