Consider the Laplace equation $\Delta u = 0$ on the below domain, with the following boundary conditions: $u=0$ along $y=0$, $u - \partial u/\partial n = 0$ on $y=-1$ and periodic boundary conditions in the $x$ direction.
This problem is ill-posed because $u=0$ is a trivial solution, but $u=y$ is also a valid solution that satisfies the PDE and the boundary conditions.
Can anything else be said about this problem? Is it possible to impose some reasonable constraint to give a unique solution? For example in the case of pure Neumann boundary conditions, one could impose that $\int_\Omega u ~d\Omega = 0$.
Note that I'm actually interested in the slightly more general case of non-flat boundaries (but with the same boundary conditions).
I think it should be noted that we can explore uniqueness properties from the perspective of an ODE. In particular, if we consider the domain $\Omega = \{ 0 \} \times [-1,0]$, we see the problem is reduced to:
$$\begin{cases} \frac{\partial^2 u}{\partial y^2 } = 0 \\ u = 0 & y = 0 \\ u = \frac{-\partial u}{\partial y} & y = -1 \end{cases}$$
Let $w = \frac{\partial u}{\partial y},$ we see $$\frac{\partial^2 u}{\partial y^2} = \frac{\partial w}{\partial y} = 0 \implies w \equiv \text{ constant }$$ $$\implies w = c \implies u = cy + d$$
Applying our initial-conditions, $$y = 0 \implies u(y= 0) = 0 \implies d = 0 $$
So we're left with the solution $$u = cx$$
In essence, the round-robin boundary condition is actually superfluous in this case. In particular, $$u(y = -1) = -c = \frac{-\partial u}{\partial y}$$
which we arrived at by the previous conditions. This results in a lack of specification for our constant $c$, which brings about our non-uniqueness. Clearly, $\Omega$ can be extended continuous to any domain $\Omega' := \mathbb{R}^n \times [-1,0]$ showing uniqueness for any dimension.
Clearly, if the ODE case is non-unique, then we see allowing $x$ to vary won't change this fact.
Of course, we could specify the average energy of our solution, as you mentioned is done for the Neumann-boundary problem. That is, we define
$$c := \frac{1}{\mu(\Omega)}\int_{\Omega} u d\mu$$
But we could also specify a boundary condition that isn't made superfluous. That is, a boundary condition that is independent of the prior conditions.
For example, if you specified the constant: $$u = \frac{-\partial u}{\partial y} \ \ \ \text{when} \ \ \ y = a \not = -1$$ then you would in fact have uniqueness as $u = 0.$