I think you could call it something that would honor the fact that most fathers actually want to do the right and fair thing by thier kids... calling it something like the "fair parenting association". I like a "father's rights" named organization also, because it puts the focus where it needs to be. You're right, "national Organization for Men" would be laughed out of existence. But "separated Parenting Association" (SPA... ok, that's a weird acronym for a bunch of men... but you see my point, look for something with a catchy acronym)... would be quite popular and would properly name the organization as one that cares about NOT try to polarize itself and embrace the extremists.
I think it needs to address court referendums, removing individual judges who show bias against men/fathers, initiating legislation that will equalize things, as well as re-framing the problem of domestic violence against men as something thats's NOT a joke... and watching media depections of men and fathers particularly as clueless buffoons who can't be trusted with kids or households or social events... It's a serious problem!
An organization dedicated to furthering the cause of proper parenting eo encourage fathers' active participation really should be embraced by all these people who whine about absentee and deadbeat Dads, and if they failed to embrace the organization, then thier bias would be clear, wouldn't it?