I've found that the type of mission doesn't matter, it's who the mission is for that counts. You need to gain rep with multiple minor factions to see the rank progression missions.
I was stuck at Lord for a long time (weeks) doing missions for empire league, emperor's grace, empire party etc etc...
In order to progress further and acquire baron I had to find a group I hadn't seen before. Even though it says I'm allied with all empire groups, the groups individual reputation towards me isn't displayed. Because of this I searched for a station owned by a local minor faction (such as a defence force, ignoring the "national" ones) and within 1 hour of mission running I acquired the next progression mission.
The galaxy map is your friend for these groups. To get started though, any imperial system will do until the progression missions start to dry up.