In general there is little interest in ‘Lindelöfizations’ per se: they don’t as a class have nice enough properties. One very specific exception (or better, class of exceptions) is the one-point Lindelöfization of an uncountable discrete space. Suppose that $D$ is an uncountable discrete space, and let $p$ be a point not in $D$. Let $X=D\cup\{p\}$, and say that $V\subseteq X$ is open iff either $p\notin V$, or $p\in V$ and $X\setminus V$ is countable. Since the countable subsets of $D$ are precisely its Lindelöf subsets, the nbhds of $p$ are precisely the complements in $X$ of the Lindelöf subsets of $D$, and the construction is analogous to that of the Alexandroff one-point compactification of a locally compact space. These spaces have proved somewhat useful, especially in constructing (counter)examples.
This construction can of course be generalized. If $Y$ is a $T_3$-space in which every point has a nbhd with Lindelöf closure, we can form a Lindelöf Hausdorff space $X$ containing $Y$ as a dense open subspace in the following way. Let $p$ be a point not in $Y$, let $X=Y\cup\{p\}$, make $Y$ an open subspace of $X$ with its original topology, and let the open nbhds of $p$ be the sets of the form $X\setminus F$ for closed, Lindelöf subsets $F$ of $Y$. It’s not hard to check that $X$ has the desired properties. I can’t offhand recall seeing it used, however.