As I mentioned in the comments, Semple and Roth (Introduction to Algebraic Geometry) discuss this in "Examples on Chapter VIII" example 12 (p.187 of my version). As far as I can tell, this is what they are saying:

Homoloidal webs:

Let $V$ be a 4-dimensional vector space and suppose we're working in
the projective 3-space $PV$ with homogeneous coordinates
$[x_0:\cdots:x_3]$. Pick four homogeneous (degree $d$) polynomials
$F_0,\ldots,F_3\in Sym^d(V)$. We get a rational map $PV\to PV$
defined by

$$[x_0:\cdots:x_3]\mapsto[F_0(x):\cdots:F_3(x)]$$

How
should we think of this? We have a 3-parameter family of surfaces
$\sum_i\lambda_iF_i=0$ (parametrised by
$[\lambda_0:\cdots:\lambda_3]\in PV^*$, called a "homoloidal web" of
surfaces) and the equation $\sum_i\lambda_iF_i(x)=0$ defines a
hyperplane in $PV^*$ consisting of surfaces which contain $x$. Our
map sends $x$ to the point in $PV$ dual to this hyperplane.

Monoids:

Now suppose that our surfaces $F_i$ all have a singularity of order
$d-1$ at the point $p$. In other words, they're all monoids in the
sense of Cayley (because they have degree $d$). They call the
corresponding birational maps monoidal transformations. Certainly $p$ will be in the base locus of this system of surfaces, so you need to blow up at $p$ to get a morphism, but I'm still not sure why this is a reasonable definition.

They do discuss examples of this immediately afterwards, but I get lost in the notation.

We define a monoidal transformation of a surface to be the operation of blowing up a single point. This new terminology is to distinguish it from the more general process of blowing up an arbitrary closed subscheme. It also goes by many other names in the literature: locally quadratic transformation, dilatation, $\sigma$-process, Hopf map, to mention a few.It would seem Hartshorne was not aware of Cayley's terminology, or at least he does not refer to it. $\endgroup$ – Carlo Beenakker Aug 13 at 19:32thinkI heard Zariski use the term “monoidal transformation” at a time when Hartshorne was still a very beginning graduate student. $\endgroup$ – Lubin Aug 13 at 21:38