How To Improvise Jeeves, feat. A Game Where You Improvise Jeeves
What Ho, World! is a storytelling game of farce and elegance that incorporates many notable elements of the Wodehouse Regency-era farces where Jeeves was a supporting character. Among them is, of course, Jeeves - I'll be using the game features of the Servant "playdeck" in order to illustrate some of the other interesting things Jeeves can bring to the table, since it sounds like you have the "relationship to master" angle pretty well down already.
One character has a habit of accidentally, teasingly, or possibly even maliciously interfering with your duties. They may spend a Spade [to send you into disarray].
Jeeves is legendarily unflappable, but it doesn't make for a very fair game if that only goes one way. It's not so much that there must be somebody who drives you to distraction, but it is important to consider the character of your relationships to your master's friends. That might mean there is someone you can't stand, but there might also be someone you fancy, someone you think is a good influence on your master, someone you think is a bad influence.
Speaking as a GM, it's much easier to set somebody up for a spotlight moment in an incidental scene when I have an idea how they already feel about the people there.
The Elaborate Errand
[Y]ou can ask the group for an errand from someone that will take all your attention. Gain a new Joker, but choose one complication and ask the other players to fill in the details.
Jeeves was all the time getting tasked with complicated stuff, but usually it played out offscreen while Bertie bumbled around in the spotlight. If your system supports extended tests of some description, like Fate contests, you can use that to frame your downtime activity, basically interspersing each part of the test between other characters' scenes.
In your example, you could be crawling the infosphere for information, which of course is visualized as a series of improbable tasks linked together by dream logic. And if you run into difficulties you can ask the other characters to provide assistance through various means they don't really understand the connection to, so when you're putting together a jigsaw puzzle made of swiss cheese but a moderator engram is guarding the last piece, you can ring the people in the ship and ask them to start a flamewar over the superiority of Blasted Flavorsqueeze to that obvious knockoff Flavorblast Squeeze, in order to distract it.
In a crisis situation you can unearth skills left fallow since you entered your current employment. Commit a Diamond and choose one field of knowledge: [...] you may demonstrate world-class competence within this field.
Admittedly this might not be relevant to the more purpose-built class of AI, but developing a skill set and a checkered past that rarely comes up is helpful in giving the GM more to hook you with than just your default schtick.
The Voice of the People
If you give someone suggestions on how to improve their attire or behavior and they accept your advice, their next Heart spend is free.
This was also a fairly common Bertie-Jeeves throughline. Bertie would adopt some ridiculous fashion or mannerism that was just the latest thing, and Jeeves would gently express his disapproval until Bertie let the thing lie.
Not that Jeeves was on the edge of fashion; rather, because he wasn't on the edge of fashion he couldn't possibly have had an ulterior motive to get Bertie to set aside his gold-plated lorgnette; it really just did look ridiculous.
If it suits you, you can play up this angle and be a moderating voice to ambitious PCs, not out of jealousy for their ambition but out of concern for its reach.
When someone comes to you with a problem and asks for your advice you may propose a simple plan of action (no more than three steps) that is guaranteed to get them the result they desire.
Alternately, you can lean in this direction. You don't need to be grasping and ambitious to brainstorm elaborate plans that are doomed to live in interesting times. You just need someone who is grasping and ambitious, and more crucially, believes in your ability to plan.
It will help if your system involves a high degree of adaptability on the GM's part, so you have some freedom to make risky and even ludicrous plans that will nonetheless pay off in the end, assuming everyone does their part.
Restoratives and Other Helpful Things
With access to a drinks cabinet, some of the powders you keep on you, a few minutes' work and the spend of a Club, you can put together a special cocktail.
Jeeves' famous restorative was quite useful to Bertie and the circles he ran in - everyone occasionally needed to recover from the consequences of a joyful night on the town or a sorrow-drenched bender.
Not that you run among drunks, necessarily. The important part here is that you cultivate a skill that is useful to everyone; an AI, for instance, could be particularly apt at fuzzing the images of people caught on camera to reduce the spread of notoriety. If everyone could potentially have a reason to need your help, you can get brought into more scenes.
Your peers maintain a confidential ledger of their masters' and mistresses' embarrassing secrets for their own amusement.
The book of secrets at the Junior Ganyemede Club was a fairly constant fixture. While I am totally suggesting that the anonymized user data collected to improve application performance is not beyond the power of your AI to de-anonymize, at some dingy little infosphere hangout tucked under the main data trunks, it might not fit the setting.
What will generally fit is that "you're just the help" and have a good set of credentials to engage in what's often called "social stealth". As long as you're not confronting or accompanied by anyone important, nobody important will think you're worth bothering with because you're not the sort of thing important people bother with.
Admittedly this is kind of dependent on the willingness of the GM to play "the help" on the opposition's side. But if they are, that opens up the possibility for scenes only you could be in, like commiserating with the battle droids in the belly of the star carrier.
Own Goals (The Good Kind)
When you're the focus your abilities are refreshed to help you push for your Goals, but everyone else can further their own Goals if they're canny and find a way.
But lastly and most importantly, if you're going to participate on the same level as the other players, you'll want to have ambitions the same as the other players. Jeeves rarely had ambitions that drove a story the way Bertie's did, but in the card game he pulls from the same Goal deck as everyone else. Not that Goals are necessarily at anyone's expense; they include entries like "cover up an embarrassing incident from your past", "get the partygoers ruining your house to leave - politely", or "score and arrange a musical". You can probably see how some of these motivations could play out, factoring in the various strengths of Jeeves.
However, your (small-g) goals don't have to be things that benefit you. They can be things you do with a mind to improve the lot of your master or your master's friends, just things that they might not value or prioritize the same way you do. As a GM, it's much, much easier to respond to a player's demand for a scene when they can clearly state what kind of progress they want out of it.