The self-coupling constant for the Higgs mechanism I have a few questions about the parameter $\lambda$ in the Higgs-Lagrangian density
$$\mathcal{L}_H=(D_\mu \phi)^\dagger D_\mu \phi + \mu^2\phi^\dagger\phi-\lambda(\phi^\dagger\phi)^2$$


*

*Is it possible to measure it directly?

*Does it absolute value have a meaning? It relates to the self-interaction of the Higgs boson (?), but can one compare this number with something else? 

*Before the mass of the Higgs was measured, did one expect a precise value for $\lambda$?

 A: Parameters in the Lagrangian determine how strong an interaction is. That is their meaning. They can be compared with other parameters of the same mass dimension (for the case of $\lambda$ it can be compared with other parameters that are dimensionless). For example in QED we have the electron-photon term,
\begin{equation}
eA_\mu \bar{\psi} \gamma^\mu \psi
\end{equation}
The parameter $e$ is dimensionless and can be compared with $\lambda$ from the self-coupling. 
The questions is measurement is a little bit subtle due to the need to renormalize our theories which I presume you are not very familiar with. If you are only doing calculation at tree-level then you can avoid this complication and you can measure the parameter directly. 
This can be done by doing $\phi \phi \rightarrow \phi \phi$ scattering  (though in practice its done in a more convenient way). The cross-section for this collision is (this is straight forward to show in QFT) 
\begin{equation}
\sigma= \frac{\lambda^2}{32\pi E^2}
\end{equation}
where $E$ is the energy of the incoming Higgs. Thus if you have a Higgs collider running at a Luminosity (a property of a collider that is typically well known), ${\cal L}$ then the number of Higgs-Higgs to Higgs-Higgs events you see per unit time is,
\begin{equation}
\frac{dN}{dt}=\frac{\lambda^2}{32\pi E^2} {\cal L}
\end{equation}
Thus by measuring number of events and knowing the energy and luminosity of your incoming beams you can extract $\lambda$.
Lastly, regarding the precise value for $\lambda$. In general it could be anything at all, but we typically expect such numbers to be within an order of magnitude of $1$ (otherwise we say a theory is unnatural). 
