As far as the options presented as a whole go:
Most Important (IE, we should have a damn good reason for not doing these before we even consider others. These are my two hard votes.)
[ ] Entrance Examination
Dragomas is forming a panel of eight Wizards from the eight Orders to determine the most suitable fit for a certain child of nobility. Be the Grey Order's representative on that panel.
This could not be more our responsibility or domain, and will have drastic implications on the future of the Empire as a whole. We're also the difference between it being a dice toss and a basically sure decision.
[ ] Dooming and Quickening
There will be rites and celebrations for Prince Mandred entering adulthood. Be present for them, and perform others that are pleasing to his true patron.
The same reasons as Entrance Examination, with slightly less ability by us to actually impact theoretical results. Taking both of these options is how we get the lay of the land for the future of arguably the most politically important person growing up in the Empire right now.
Probably Important (It would probably be good to do these, but mostly for situational awareness of ongoing events of worldly importance)
[ ] Skull River Ambush
Look into the investigation of the mining of the Skull River, and any consequences of it.
It's probably good to know who specifically we should be expecting to deal with re: Marienburg opposition.
[ ] Kalashiniviks
Investigate the fate of the Kalashiniviks, who were made a scapegoat for the death of the Tzar.
While there is no practical way anyone will ever know we assassinated the last Tzar, shit could turn really poor in Kislev if the Kalashiniviks were able to credibly prove it wasn't them and turn the death of the Tzar into a still unresolved mystery.
[ ] Druchii Diplomats
Check in on these unexpected visitors to Tor Lithanel.
Why the fuck are these guys here all of a sudden? What do they know? I, personally, do not like them being here right when the Waystone project suddenly starts paying dividends.
Most In-Character (Things that would absolutely be central to Mathilde's character. 'Mathilde Interrupt', if you will)
[ ] Reading on Nehekhara
You recently acquired a great many Dwarven books on Nehekhara, including ones written in the direct aftermath of the Great Ritual by Dwarves who had known the Nehekharans when they still lived. It's unlikely your official duties will ever require you to delve too deeply into the topic, so the only way to indulge your curiosity is to dedicate some of your leisure time to the task.
Honestly, people in this thread have sold me pretty well on continuing to dig regarding Nehekharan stuff. I think it's absolutely in Mathilde's bag to sort of semi-accidentally be assembling one of the most complete educations on Warhammer necromantic history outside of Heinrich Kemmler himself.
[ ] Initiate
For the first time, a child born in Karag Nar has shown a capability of using magic. It is the duty of a Wizard to ascertain their suitability to join one of the Orders of Magic.
She may not be Loremaster anymore, but this is Mathilde full circle - she's created a place where she can be the person who wasn't there for her before Regimand showed up.
[ ] Witch Hunter
There's apparently some sort of Witch Hunter snooping around Gretel's lands. She'd absolutely be capable of dealing with it, but you'd be able to do so faster and would have more fun doing it.
Any time Mathilde ever willingly passes up an opportunity to mog a Sigmarite, I scratch one big tally in the 'suborned by Tzeentch' log.
[ ] Orb Reveal
If you don't feel like waiting until you finish writing the book on how, you can make your grand reveal of the Orbs of Sorcery you created to the Colleges now.
The only thing Mathilde likes more than dropping inexplicable lore is impossible achievements. I'll also reiterate that this would combo fantastically if we really wanted to walk into the Entrance Examination option and make it clear who the fuck is calling the shots on where our godson is going. God knows the Greys are already going to look prescient once it gets out that we already had a Magister as his godmother from the get. On the flip side, we are going to give them over eventually anyway.
Absolutely Not (I'm not saying any of this is bad, it's just unjustifiable IMO in the face of the other options)
[ ] Amber College
Check in on the salamanders.
If 'this could have been an email' was made an option.
[ ] Swordplay
Test your novel swordfighting style against the swordsmen of Tor Lithanel.
Come on now. We're already basically instantly lethal to any mundane physical opponent. Boney as much as said that we were about to (or already) hitting diminishing returns in regards to sword-mastery; frankly, if we run into an extended duel with anyone we only could have defeated with something better than the invisible Zweihander iaijutsu we already have, we have fucked up so horrendously they should take away our Lady Magister, Grey College and wizard licenses in that order.
Fucking Awesome
[ ] Okri
You've met Loremaster Okri of Karak Eight Peaks once before. Pay him a visit and see how his great ambitions for heavily-armed Ironbreakers delivered by Gyrocarriage are going.
Anyone who doesn't vote for giving Belegar his own unit of Dwarven Helldivers is going in the fucking Book on principle.
Ambivalent (Could do it or not do it, no strong opinion)
[] Everything Else