Removing alignment entirely is a possibility. Or, perhaps a less-intrusive goal would be to remove alignment from all things that don't have "special alignment auras." In that case, one must ask "what do we do about those mechanics which depend on it?"
The important effects are... lessee... Protection from X, Detect X, and the Word line. If there are any important ones I'm forgetting, lemme know.
- Alignment restrictions for acquiring features, generally speaking, can go the way of the dodo without impacting much of anything. Paladins no longer have alignment requirements but do have codes of conduct. I recommend defining new codes of conduct rather than using the existing ones, since queries like "evil acts" return "undefined."
- Protection from X spell could perhaps become reflavored as a sort of "Planar / Possession Ward" spell. It prevents summoned extraplanar creatures from entering an area, it provides an AC bonus against anything not native to the caster's plane or summoned, undead, clerics, paladins, and it suppresses attempts to control you (compulsions and such). Maybe make it interact with Eberron's possession rules. The most important part of PfX is the suppression of compulsions, so it's important to keep that part intact, and that seems to fit with the "possession ward" flavor.
- Detect X becomes "detect clerics/paladins/blackguards/outsiders/undead/deathless" or more simply "detect aligned auras / aligned energy channeling." Clerics ping as the alignment of their deities, regardless of their own personalities. Paladins / Blackguards / etc all ping as the alignment of their aura. Deathless always ping Good, other undead always ping Evil, Outsiders ping of their respective alignments. Anyone who doesn't have a deity (or has a neutral aligned deity) who can turn undead pings Good, and vice versa for Rebuke Undead.
- Holy Word / Blasphemy / Dictum / Word of Chaos simply affect all beings not from your plane that don't share your deity's aligned aura. If a balor does it on the material plane, he's hitting everyone except for, say, clerics of Erythnul.
- Radiant Charge / Doom Charge / Tide of Chaos / Law Bearer can simply work on any sentient enemies whom you have a specific moral disagreement with (e.g. you must know something, even if it is a minor or generalized detail, about the target. You can't use it on random peasants). Radiant charge may work on undead/evil outsiders, Doom charge may work on Deathless/good outsiders. To qualify to have these maneuvers, your deity's alignment must be within one step of the maneuver's alignment requirement.
- Aura of Chaos / Perfect Order / Tyranny / Triumph stances make you ping as if you were a cleric of a god of those alignments. Aura of Triumph simply procs against all targets.
- Smite works on everything. No I don't care that this is a buff, Paladins are a low tier class.