Primogeniture of Imperial High Noble titles is the usual thing, but its not a sure thing. Along about the time a potential heir reaches the age of majority, he is sent on a Grand Tour of his future realm and its neighboring areas (at least). Officially, this is done to familiarize them with the taxation districts which they may one day administer for the Emperor. Unofficially, it's done to allow the most important of their future subordinates and allies (and their assigns) a chance to check him over.
If a particular heir passes muster with each individual visited, only the usual polite noises are made and this is taken by the parent as conditional approval. But if a particular heir proves objectionable to someone for any reason, the polite noises are supplemented by a substantive missive sent by discrete courier. The fact that someone voiced concern is a serious matter, requiring delicate negotiations and prompt remedial actions. If more than one person voices a concern - particularly if its the same concern - this usually means that particular potential heir will likely "decide" to pursue a career that is incompatible with remaining the heir to the noble seat. If none of the potential heirs are found to be acceptable by all concerned, then the search for an acceptable heir widens out to include lesser branches of the House. Once the various Grand Tours are complete, the Heir Presumptive is confirmed and his/her intense training begins.
Of course, that's the way it's supposed to work. Perhaps surprisingly, it usually does work that way. But every now and again, a delusional High Noble decides that his "unacceptable" heir is a special exception and forges ahead with the announcement of his/her confirmation. This quickly gets embarrassing for all concerned, as the people who objected know their concerns have not been properly addressed and so pass them on up over said High Noble's head. Said High Noble then receives a gentle, but pointed, inquiry from his/her higher-up, asking if the noble in question has really properly considered his actions.
A group of PCs who have gained a reputation as discreet troubleshooters are hired as escorts for a Registered Companion* while she carries out a long-term engagement at the Subsector Court. The PCs are not told that she's there as a direct - but unofficial - representative of the Sector Duke, tasked with persuading the Subsector Duke to select someone other than his spoiled-brat eldest son as his Heir Presumptive.
- The spoiled-brat - not understanding the realities of polite society - decides that he wants the Companion for himself. The PCs must successfully dissuade him without resorting to violence.
- The Subsector Duke knows very well that his son is a rotten apple. But he's being blackmailed by someone who wants his successor to be a weak and easily-manipulated figurehead. So the Companion - no matter how skilled - will get nowhere. She will feel steadily growing frustration. Eventually, with no other course of action evident, she will confide in the PCs and seek their aid in discovering the source of the Duke's intransigence.
- The "someone" who's blackmailing the Duke realizes what the Companion's mission must be and reacts by attempting to lure her into a situation where she will appear to violate her ethical code. If they discover that she has confided in the PCs, that will be sufficient (assuming they can get proof of it). They will then attempt to blackmail her into delivering a favorable report on the rotten apple to the Sector Duke.
- The PCs discover the identity of the blackmailer . . . and it's the local patriarch of the Church Of Stellar Divinity, who is hoping to use the future scandalous Duke as an object to be vilified and to lead a traditionalist backlash against. His long range plan (if discovered) is to use his "demonstrated leadership qualities" to be appointed as the new Duke . . . then use his position to further the influence of his faith in the Subsector.
*The Registered Companion is borrowed from the Firefly series, and is an individual who in this case serves as both courtesan and "back channel" envoy. They implicitly have strong ethical codes concerning their assignations and associates. A procedure for generating characters to fit this position can be found as "Paid Companion" in Ken Murphy's character generation article, The Invisible Masses. A plain-vanilla diplomat can be used instead, if for some reason a Registered Companion is deemed unsuitable for the campaign.