Edit, 2/25/15: Well, after discussing it with Nerses in the comments, it seems I was wrong; taking homotopy fixed points does work! (If you're already guaranteed that the space is a free loop space, that is.) The argument is at the end. Most of my original answer is below.
First, there is a general machine you can try to run in situations like this to see what structure is available: in any category with finite coproducts, the functor $\text{Hom}(c, -)$ naturally acquires the structure of a model of the Lawvere theory whose $n$-ary operations are given by maps $c \to nc$, where
$$nc = \bigsqcup_{i=1}^n c$$
denotes the coproduct of $n$ copies of $c$. This is because $\text{Hom}(nc, -) \cong \text{Hom}(c, -)^n$, and by the Yoneda lemma, natural transformations $\text{Hom}(c, -)^n \to \text{Hom}(c, -)$ are naturally in bijection with homomorphisms $c \to nc$.
If you run this machine in the homotopy category of pointed spaces with $S^1$ the circle, you get that $\text{Hom}(S^1, -)$ naturally acquires the structure of a model of a Lawvere theory which turns out to be the Lawvere theory of groups, as expected. Similarly, for $S^n, n \ge 2$ you get the Lawvere theory of abelian groups. (Details here; other examples here and here.)
Unfortunately, any version of this machine in unpointed spaces suffers from the same problem: in unpointed spaces, the coproduct is the disjoint union, so e.g. morphisms from a circle $S^1$ into a disjoint union $n S^1$ of $n$ circles factor through the inclusion of one of the connected components. This means that the resulting Lawvere theory has no interesting $n$-ary operations, and that everything is determined by the unary operations, or by morphisms $S^1 \to S^1$.
And this is just not a lot of data. Consider the case that $Y = BG$ is the classifying space of a discrete group $G$. Then $LY = LBG$ is the classifying space of the groupoid $G/G$, the adjoint quotient of $G$, which has objects the elements of the group and morphisms given by conjugation. This is equivalent to the disjoint union of the classifying spaces $B C_G(g)$ of the centralizers of a representative of each conjugacy class of $G$. A map $S^1 \to S^1$ of degree $k$ sends the conjugacy class of $g \in G$ to the conjugacy class of $g^k$, but otherwise the different conjugacy classes don't interact, so I don't see any hope for gluing them back together via a procedure like taking homotopy fixed points.
Edit, 2/25/15: Now for the argument. I was being silly. It's very straightforward:
Theorem: Let $S$ be a topological group and let $Y$ be a space. Then the space of $S$-homotopy fixed points for the natural action of $S$ on the mapping space $[S, Y]$ is weakly equivalent to $Y$.
Sketch. By the universal property, $S$-homotopy fixed points for the natural action of $S$ on $[S, Y]$ is weakly equivalent to the mapping space $[S/S, Y]$, where $S/S$ now denotes the homotopy quotient of the natural action of $S$ on itself by translation (not conjugation). But $S/S$ is a point. $\Box$
But because I was so confused about the case $S = S^1, Y = BG$ I want to go through it in detail. First let's describe explicitly what the action of $S^1$ on
$$LBG = B(G/G) = \bigsqcup_{[g]} B C_G(g)$$
is, where the union runs over all conjugacy classes $[g]$ of $G$. One of my confusions is that I forgot that taking $g$ to be the identity gave a copy of $BG$ sitting inside $LBG$ to aim for; what happens when we take homotopy fixed points is that all of the other connected components just disappear, rather than becoming spliced back together.
Giving an action of $S^1$ on each component amounts to writing down a map $S^1 \to \text{Aut}(B C_G(g))$ for each $[g]$, hence (after taking $\pi_1$) writing down a map $\mathbb{Z} \to Z(C_G(g))$. Of course there is only one natural candidate for such a map, which is the map sending a generator $1 \in \mathbb{Z}$ to $g \in Z(C_G(g))$. The action of $S^1$ on $B C_G(g)$ obtained in this way is encoded in a fiber sequence
$$B C_G(g) \to E \to B S^1$$
where $E$ is the homotopy quotient of $B C_G(g)$ by this action, and the homotopy fixed points of the action are homotopy sections of the map $E \to BS^1$. The long exact sequence in homotopy applied to this fiber sequence is
$$1 \to \pi_2(E) \to \pi_2(BS^1) \cong \mathbb{Z} \xrightarrow{g} C_G(g) \to \pi_1(E) \to 1$$
and we conclude that $\pi_2(E) \cong |g| \mathbb{Z}$ if $g$ has finite order $|g|$ in $C_G(g)$ and that $\pi_2(E) \cong 0$ if $g$ has infinite order. It follows that the map $\pi_2(E) \to \pi_2(BS^1)$ has no homotopy sections unless $g$ is the identity, so in fact only the conjugacy class of the identity contributes to homotopy fixed points. In this case the action of $S^1$ is trivial, $E = BG \times BS^1$, and hence the space of homotopy sections is the space of maps $BS^1 \to BG$. But since $\pi_1(BS^1)$ vanishes this is just $BG$ itself.
