Any elements that can be connected to the cavity under a stud? Are there any elements that can be connected to the 'cavity' (i.e. the space) under a stud (e.g. on a regular plate or a regular brick)? The cavity measures about 2.4mm in diameter. 

So far I've only tried a minifig hand peg (not the clip side, the other side) but unfortunately it is just a bit smaller, it has a diameter of about 2.3mm.

[Edit] I'm talking about the cylindrical cavity with a diameter of 1.5 LU as indicated in the overview posted here -- What are the dimensions of a LEGO brick?.
If I'm understanding you correctly, you're talking about the molding cavity directly underneath the stud (that actually goes up into the stud), and not the main tube cavities under plates/bricks (that are actually offset from the studs).

If that's the case, no, I'm not aware of any element that "intentionally" fits that space.

First off, I don't think that space is made with a high tolerance.  Looking at some of my bricks, the dimensions of that cavity are not nearly as uniform as the other features of the brick.

Secondly, I don't know how many System bricks have components that are not integer multiples of LUs.

You might get lucky and get a minifigure piece that has a detail that is 1.5 LUs, but you'd have to try a lot of them.