You just need to lobby the right people. Starting with any of your realm mates who are full members is a good bet. Getting elected in BM is just like getting elected in the real world; you have to put yourself out there, press the flesh, and lobby as many individual people as you can for their votes. It also doesn't hurt to get people with influence to endorse you.
You'll also want a platform. "Here's what I stand for, and what I'll do if I'm elected" is a good start. "Here's what's wrong with the system and here's what I intend to do to make it better" is another viable approach. Eoghan actually probably has a decent shot at election. He's known to most of the membership at this point and he's generally taken the high road on the issues he weighs in on. Nothing to drag down your reputation, and plenty of evidence that he's a reasonable, well-spoken individual.
Sage advice. Alaron, in contrast to Eoghan, is more of a maverick (I hate using that word, but it applies here). He is slowly finding his voice and becoming more active and vocal, as can be seen. For example, the pilgrimage to D'Hara, and of course the whole thing about the Magistratum. Given what he's hearing from Allison now, I wonder whether standing on a reform and/or anti-corruption platform would do him any favours. However, he is slightly unstable - the whole thing about removing foreign priests with the dagger probably does him no credit, even when he tried to frame it as being his prerogative as a Lord to allow or forbid preaching as he saw fit.
As far as campaigning.. well, he's sort of well-known within his army and usually volunteers for special assignments, so that may provide some backing. Also been a Lord for most of his time in Astrum, helping to rebuild his region. So at least he has evidence of taking action to fix things. Still might be difficult though. But it certainly will be fun.
On that note, what would it take to secure Lysander's backing? Name your price.
