Researching people on LinkedIn isn't weird - that's the point
Ahead of an initial meeting with (i) new prospects or (ii) new members of an existing client's team, we all should be doing prep work, including looking-up those new contacts on LinkedIn.
Often it's just for general background info, but sometimes you discover you’re a 2nd degree / share a connection, which can be hugely helpful in building that new relationship – (assuming it's a real connection, not a recruiter from 18 months ago, etc.).
Yet for some reason lots of people feel it's creepy or weird to mention the common connection because it implies you looked-up the person on LinkedIn ahead of time.
It's only weird if you make it weird – (this isn’t LinkedIn creeping ahead of a first date).
It's assumed that a diligent professional would do that type of pre-meeting prep.
That said, it makes sense to wait until the end of the meeting to mention it, once you've built a bit of rapport, but you should certainly make the common connection explicitly known.
As the meeting is ending, float a quick, "Hey, by the way, ahead of the meeting I looked-up your profile on LinkedIn, and noticed you're also connected to Person X. How do you know them?" etc.
It's not weird, and can go a long way in creating a real connection with the person.