  For a fixed $\alpha$, each real number $x \in (0,1)$ can be represented by
many different generalised $\alpha$-L\"uroth expansions. Each such expansion
produces for the number $x$ a sequence of rational approximations
$(\frac{p_n}{q_n})_{n \ge 1}$. In this paper we study the corresponding
approximation coefficients $(\theta_n(x))_{n \ge 1}$, which are given by \[
\theta_n (x): = q_n \left|x-\frac{p_n}{q_n}\right|.\] We give the cumulative
distribution function and the expected average value of the $\theta_n$ and we
identify which generalised $\alpha$-L\"uroth expansion gives the best
approximation properties. We also analyse the structure of the set $\mathcal
M_\alpha$ of possible values that the expected average value of $\theta_n$ can
take, thus answering a question from
\cite{Barrionuevo-Burton-Dajani-Kraaikamp-1994}.
