The least such ordinal $\beta$, often written $\beta_0$, for which $L_\beta$ is a $ZF^-$ is also characterised as the "ordinal of ramified analysis". This is because the ramified analytical hierarchy, which builds up cumulative second order number theoretic structures of the form $$\underline{P_\alpha}=( P_\alpha,\mathbb{N}, +, \times, \ldots)$$  with $P_\alpha\subseteq P_{\alpha +1}\subseteq
\mathcal{P}(\mathbb{N})$ by looking at all sets of integers definable over $\underline{P_\alpha}$ by using instances of second order comprehensionto to obtain $P_{\alpha+1}$.  (Unions are taken at limits). This hierarchy has height exactly $\beta_0$ (meaning $P_{\beta_0}=P_{\beta_0+1}$.)   Alternatively put: $\underline{P_{\beta_0}}$ is then the least model of $Z_2$, or full second order comprehension (sometimes just abbreviated to "analysis", or ``second order number theory'').

$\beta_0$ of course is $\Sigma_n$-admissible for all $n$. $L_{\beta_0+1}$ sees that $\beta_0$ is countable, and so I suppose one may define a particular $\omega$ sequence in $L_{\beta_0+1}$ cofinal in it by just taking the $<_L$- least such.

One can also paraphrase the assertion that $L_{\beta_0}$ is the least $\beta$ with $L_\beta$ a $ZF^-$ model, as being the least such that for no $n$ is the fine-structural projectum $\rho^n_\beta$ less than $\beta$. (But this is really only dressing up one concept in the jargon of another.)

$L_{\beta_0}$ is (I believe) the least level of the $L$ hierarchy whose reals form a model of $\Delta^1_4$-Determinacy (Martin - unpublished).

For the Ramified Analytical Hierarchy: see Boyd, Hensel, Putnam "An intrinsic characterisation of the ramified analytical hierarchy", JSL, late 60's I believe.   This hierarchy is not much studied these days, but was of interest in the pre-Jensen fine-structural era of the analysis of levels of $L$ (the latter's work now surpassing it).
For connections between models of subsystems of second order comprehension and constructible sets, see also S. Simpson's book "Subsystems of second order arithmetic".