Try to cast a helpful Divination spell on them, like Guidance
Guidance is a divination cantrip, so Mind Blank would make them immune to its helpful effects. If you try to cast Guidance on someone and it fails, you'd realize that because you wouldn't be concentrating on a spell. (Normally it's concentration for 1 minute, or ends early if the target chooses to add +1d4 to an ability check)
Or a harmful one like True Strike or Hunter's Mark.
We can add this to this short list of use-cases for True Strike :P
Finding a target immune to one of those spells would leave mind blank near the top of a short list of possibilities, depending on what else you know about the creature.
I don't have a rule citation for exactly what happens if you try to cast a spell on a creature that's immune to spells of that school in general. I'm assuming it fizzles, so its duration doesn't start and you aren't concentrating on it.
Cast something that should be visible with Detect Magic, but isn't
Mind Blank stops Detect Magic from detecting any magic affecting the target, as the creature is immune to that divination spell. If you do something that should leave a detectable aura of magic on the target, but it doesn't, they might have Mind Blank up. But not Intellect Fortress; that doesn't block divination.
(Or Nystul's Magic Aura with the False Aura effect, if that works on creatures as well as objects. With a strict reading it only works on objects. Even if it does work on creatures, you should still be suspicious if a recent ally of convenience is using Nystul's to hide something, if you don't know why they'd be doing that. Still, you couldn't then assume Mind Blank specifically.)
Some other anti-divination effects like Nondetection stop the creature from being targeted by "divination magic", rather than making them immune to all "divination spells". (The target can't be targeted by any divination magic or perceived through magical scrying sensors.) Thomas Markov's answer argues that Detect Magic (even though range=self), creates an AoE effect that "targets" creatures or objects you're looking at. If your table uses that ruling, then Nondetection is another explanation for not being able to see magic on them.
So for example, if the party needs to use Fly or Spider Climb on one member so they can carry a rope (or the other members 1 at a time) over some terrain, select the party member you don't fully trust and buff them up (hrm...). Have someone else use Detect Magic while you're concentrating on that helpful spell. (Detect Magic is also concentration.)
Or while they're taking a nap, cast Faerie Fire on them. Or just "as a joke". It's non-damaging, but doesn't require a willing target. (And it's a Dex save to avoid, so they auto-fail if actually napping.) Faerie Fire is maybe not ideal because you (or the DM) could argue that it's a visible magical effect stuck to the target, not intrinsic to the target's body, and thus Detect Magic could see that it's magic regardless of Mind Blank.
Many other non-damaging effects are charms, like Charm Person, and thus blocked by Mind Blank. (The Friends cantrip requires you to pick a creature, but has Range = self. It's of the enchantment school but doesn't mention creating the charmed condition. It's debatable whether it leaves an aura of magic on the target while active.)
The Message cantrip is Transmutation, and has a duration of 1 Round. (Not concentration but not instantaneous). Arguably it should leave an aura of transmutation magic around the recipient while it's being cast, so you can have Detect Magic up while you send a Message to someone.
The Resistance cantrip is Abjuration, so unlike Guidance it will take effect.
Prestidigitation would take some rule-bending to use on a creature: You make a color, a small mark, or a symbol appear on an object or a surface for 1 hour. - A creature's skin may not count as "a surface". And even then, it might just be an external effect, like I mentioned above for Faerie Fire.
Or convince them to cast anything on themselves.
In combat, with detect magic up:
Many effects, such as Slow, are non-charm and should show up with Detect Magic. (But wouldn't if the target has Mind Blank.)
Bane is an enchantment spell but doesn't cause the Charmed condition.
Hunter's Mark is divination, and will fail on a creature with Mind Blank. (Hex is enchantment.)
Find some divination magic that isn't a spell?
Mind Blank specifically says you're immune to "divination spells", different from Nondetection protecting you from divination magic in general. (The target can't be targeted by any divination magic or perceived through magical scrying sensors.)
Nondetection would block include creature or item abilities to sense magic, but Mind Blank wouldn't. Also non-spell class abilities if there are any that detect magic.