7:58 PM
My idea ultimately was that every Bousfield class has the property that $\langle E\rangle\geq \langle 0\rangle$, and so either its minimal (i.e. there is nothing between it and $\langle E\rangle$) or it factors through a minimal Bousfield class.
Then, I was thinking that there was a classification of the minimal Bousfield classes away from $\langle HF_p\rangle$, but now I'm not so sure that's true.
@TimCampion I'm also not familiar with any special name for the Bousfield class of a ring spectrum. They are not unique in being idempotent.
One thing that seems to provide a lot of counter-examples is the Bousfield class of the Brown-Comenetz dual of the sphere, $\langle I\rangle$, so it might be worth seeing if you can figure out (or if someone else knows) what $L_I(\mathbb{C}P^\infty)$ looks like.
It's conjectured that $\langle I\rangle$ is minimal as well. Also, if the $A(n)$ spectra are non-trivial (i.e. the telescope conjecture fails) then their Bousfield classes are also minimal.
So anyway, I think this is all just getting us back to Eric's declaration that weird stuff happens in the Bousfield lattice.
My intuitive picture of the Bousfield lattice is that it has minima at all the $\langle K(n)\rangle$, and then it has this kind of "trap door" at $\langle HF_p\rangle$ below which all hell breaks loose (and this is where $\langle I\rangle$ lives).