Conventional ethical relativism appeals to some set of social norms, traditions or practices, or personal convictions as truth-makers or grounds for moral justification. On this model, for any moral precept p, the statement 'p is true' means something like 'I (and/or those of my social group) accept p' -- where such acceptance in the assertor's moral theory (perhaps linked with some anthropological and psychological theses) constitutes the relevant "truth" or justification of p.
It's striking that conventional relativism has so uniformly taken something like this form, for there's no reason to assume that a "relativistic" morality should be relative only (or even primarily) to such positive (i.e., man-made) conventions or precepts.  Why not instead suppose that moral precepts are relative, say, to certain innate objective psychological characteristics -- motivations, dispositions, potentials? On this approach, "p is true" would mean something like 'By dint of certain objective psychological facts, p is true as applied to me (and agents relevantly like me)" -- where 'true' is analyzed in terms of some more or less standard, nonrelativist notion of truth (coherence, justified true belief, a function of an epistemically reliable process, whatever) and truth value is cashed out, say, in terms of the agent's flourishing. 
Call this sort of ethical relativism Reformed Relativism ("RR"). RR has some obvious advantages. It construes moral precepts as functions that play out in the natural world; as such, it provides a relatively clear-cut criterion for their verification or justification: A moral proposition is "true" or justified relative to an agent A just in the case it commends an activity or practice that is conducive to A's flourishing. It also sits well with the common notion that there are different "personality types," the existence of which suggests that there might be different "moral types" as well (moral character being a feature of personality). As well, it is consistent with the idea that an agent might be mistaken about the standards he takes to be his own (because the standards obtain not by dint of a conscious, positive commitment to them, by by dint of deep, perhaps subconscious psychological traits, such that epistemic access to them isn't guaranteed) and, similarly, with the notion that an agent is sometimes bound by principles that rub against his transient inclinations or affects.
On the other hand, RR might do nothing to quell moral disagreement, and in fact might just make it worse, since at least prima facie such a reformed relativism increases the possible number of conflicting moralities from the number of possible group-based moralities to the number of individual-based moralities (a la garden variety subjectivism). It also might make the "correct" morality cognitively inaccessible, the moral instincts involved being inherently more numinous or inchoate than consciously adopted norms; without significant, possibly invasive examination into the individual's psychology, then, it's possible that RR would leave us unable to tell whether in giving a moral justification an agent is being honest with us, and indeed leave him unable to tell whether he is being honest with himself. And of course given the numinous quality of the agent's "deep" character, there is also bound to be uncertainty about how actions or practices causally link to his flourishing, as indeed there will be uncertainty about what his flourishing would (or should) consist in.
In any case, over the next few weeks (months?), I'll be exploring these and related ideas. That's all for now...
1. Other forms of relativity do not share this positivistic aspect. In what I'll call "w-relativity," for instance, a genetic trait is fitness-enhancing relative to an environmental milieu. In certain forms of semantic relativity, a modifier is relative to a class of the object it modifies (e.g., a *big* shrimp is smaller than a *tiny* elephant). In Bayesian probability, the soundness of an inference is relative to its prior distribution (or knowledge, or probability). And of course, in the classic example of Einsteinian relativity, the truth of a physical measurement is relative to an inertial system or "frame of reference." In none of these cases does truth or justification turn on the objectively unconstrained, positive adoption of some underlying principle peculiar to the relevant relativised parameter.
2. To be sure, sophisticated relativists do appeal to objective psychological properties in grounding the truth or justification of moral norms, but I think they have something slightly different in mind when they do so. They are concerned with the general traits that "morality" properly conceived must accommodate; variations outside these dimensions can then be shown to be justified. Whereas what I'm suggesting is that such traits might for a particular agent actually entail (and not merely justify or permit) the variation. To take a fanciful example, supererogatory acts are generally viewed as permissible rather than obligatory; but perhaps some persons are so morally exceptional that in fact they (but not others) would be obliged to act in a "supererogatory" manner.
Some follow-up posts on this topic: