Does this dimensioneless quantity have a name? When studying creeping flows, a common choice for a characteristic pressure scale is $$p_0 = \frac{\mu_0 U_0}{L_0},$$ where $\mu_0$ is a reference dynamic viscosity, $U_0$ is a reference velocity and $L_0$ is a reference length.
This leads me to think about the dimensionless quantity $$\frac{pL}{\mu U}.$$ Does it (or the inverse of it) have a specific name?
 A: I am unaware of any dimensionless quantity that this represents, and Wikipedia seems to agree. However, what you have defined there is the ratio of the pressure stress to the viscous stress, and this is in some sense similar to the Bingham Number (yield stress to viscous stress).
So what would your number mean? Let's go ahead and call it the Toliveira number for the time being. When $To \gg 1$, the pressure stress is much more important than the viscous stress. This means that dilation is a bigger effect than shear (remember, pressure stress is the trace of the stress tensor). So a fluid packet is growing or shrinking isotropically much more than it is deforming under shear. 
The inverse of this, $To \ll 1$ implies that the volume of the fluid element is not changing nearly as much as the shape of the fluid element is changing. Shear dominates dilation. 
I could imagine this number maybe being important defining regimes where volumetric heat release is important (combustion for example) or regions where compression is large but viscous forces are small (shocks). But I have a feeling there are numbers more directly on-point in those cases. I'll keep digging.

After some more digging, I did find one number that is kind of close. The Poiseuille Number is defined as:
$$ P = -\frac{d p}{dx} \frac{L^2}{2\mu U} $$
This relates the pressure gradient in a laminar duct to the viscous forces in the duct. For an incompressible flow, the pressure doesn't mean much anymore but the pressure gradient is important as this is what drives flows. In your comment, you asked if the number you gave has any meaning in an incompressible flow. I am fairly certain the answer is no because absolute pressure is not what matters, but pressure gradients do matter. 
A: One way to think about the Reynolds number is to note that is measures the relative magnitude of viscous and ideal (inertial) stresses in the fluid
$$
{\it Re}^{-1} = \frac{\mu\nabla u}{\rho u^2} = \frac{\mu}{\rho Lu}\, .
$$
However, the stress tensor of an ideal fluid has two terms, one related to pressure, and one related to inertial stresses
$$
\Pi_{ij}=\delta_{ij}P+\rho u_i u_j\, . 
$$
The relative importance of these two terms (in a compressible fluid) is governed by the Mach number of the flow
$$
{\it Ma} = \frac{u}{c_s} = \sqrt{u^2\left(\frac{\partial P}{\partial \rho}\right)^{-1}}\, . 
$$
Your dimensionless parameter measures the relative importance of viscous stress and ideal (pressure) stresses
$$
{\it New} = {\it Ma}^{-2}\cdot{\it Re}\, . 
$$
This is the expansion parameter of the hydrodynamic expansion (the justification for dropping terms beyond the Navier Stokes term) in compressible fluids. This is not as exotic as it sounds, cold atomic gases, nuclear liquids, quark gluon plasmas, an many problems in ordinary gas dynamics are in this category. ${\it New}$ does not have a name, but it deserves one.
A: In any dimensionless formulation of the Navier-Stokes equations you will never find a dimensionless number as a coefficient infront of the dimensionless pressure gradient. This is because the pressure gradient must always be of the same order as one of the other viscous or inertial terms:
$$\boldsymbol{v}\cdot\boldsymbol{\nabla}\boldsymbol{v} = -\boldsymbol{\nabla}p + \frac{1}{\mathrm{Re}}\Delta\boldsymbol{v}$$
$$\mathrm{Re}\boldsymbol{v}\cdot\boldsymbol{\nabla}\boldsymbol{v} = -\boldsymbol{\nabla}p + \Delta\boldsymbol{v}$$
For this reason depending on the regime (viscous or inertial) we are in, the pressure scale changes:
$$p_{vis}=\mu\frac{U}{L} \quad p_{in}=\rho U^2$$
Now it is always possible to rearrange variables to generate a dimensionless 'quantity' like you have:
$$\Pi=\frac{pL}{\mu U}$$
but i would not consider this a dimensionless 'number' as it doesn't characterize the dynamics in anyway like e.g. the Reynold's number does. That is the magnitude of $\Pi$ gives no information about the terms in the Navier-Stokes equations. Instead it is simply a ratio of quantities (aka simplex) much like $x/L$ is a ratio of two lengths.
