As far as personality traits go, I'd like to avoid sycophantic always-supportive types who are supposed to be canonical buddies. ME was a bit of a worse offender, between Liara and Tali and Garrus, and that's on top of the general hero worship from the sympathetic cast.
I'm also getting tired of deadpan snarker- they're good occasionally, but they seem to creep more and more into every Bioware character until they've infected half the cast for the sake of in-jokes and witty one-liners. Let's get more sass and tear-downs like Vivienne.
I'm also tired of 'dark and edgier' characters being either outright woobies (Jack), or the 'grizzled nigh-amoral veterans' like Zaeed, Wrex, and even (somehow) Grunt.
I suppose some things I'd like to see...
-A utilitarian moralist.
In most Bioware games, the 'good' characters, the ones who openly let morality guide them and chide the player on ethics, are almost invariably empathy-based immediate-consequentialists (with a touch of deontologist). I'd like to see a 'greater good' moralist- someone who judges and chides the player not on the immediate impacts, but on expected consequences and greater good issues: the sort who would certainly approve of saving someone, but lambast the player for things like letting Balak get away.
Paragon-ism was dealt with a remarkably soft hand and generally treated as the 'unquestionably moral' avenue. I'd like a companion who can soundly challenge the ethics of being too nice.
Continuing on this thought, another character as a foil/contrast to the utilitarian moralist-
A cost-aware deontologist. Or, in other words- 'morality is worth other people dying for.'
In most Bioware games, deontologists (the idea that ethics are inherent in the action, not the consequence: the ends don't justify the means) are paraded with the paragon ideology and pretty much never address or are forced to address the (potential) consequences of their choices. Shepard can go 'I won't let fear compromise who I am' because, hypocritical counter-examples aside, Paragonism never actually fails. Consequentialism in Mass Effect, despite being rhetorically supported by the Renegade, often supports the Paragon because of meta-gaming knowledge that Paragon options routinely offer the 'best' results with the most lives saved. Let Balak go to save a dozen workers, and the terrorist who tried to kill millions is never indicated to have killed any other humans. Players (and companions) get to focus on lives saved, and don't really dwell on costs risked or incurred.
I'd really like that sanctimonious 'morality of the moment' to give way to someone who is aware and dwells on consequences... but chooses non-consequentialist morality anyway. Rather than fair-weather deontologists who are really ambivalent consequentialists (they prefer the right thing, but go along with consequence-driven actions whenever obvious or immediate enough), someone who would take deontology to it's logical conclusion: not just being willing to self-sacrifice, but being willing to sacrifice others as a consequence of moral actions.
This isn't, to be clear, fanaticism or zealotry in which the suffering of others is cast as being a moral obligation. This isn't even a hypocritical 'I'd sacrifice others but not myself.' This is action-based morality even with the acknowledgement of negative consequences that may, can, and do result, on the basis that the consequences born by others doesn't change the morality of the act... even if the character grieves and mournes for the losses.
Such a character wouldn't necessarily be popular. They'd probably be hated by the people who find them immoral. But they'd be a good foil for the occasionally sanctimonious aspects of paragonism, which I think is called for in a ME reboot, just as the consequentialist moralist would be a good change for the 'morality-driven' archetypes.