There is, sort of, and the idea was developed in the paper "Induced fibrations and cofibrations" by Tudor Ganea in a 1967 paper appearing in the transactions.

The idea is roughly this:
given a cofibration
$$
A \to X \to C ,
$$
let us set $X_1 = \text{hofiber}(X \to C)$, where the latter means the homotopy fiber.
Then there is a preferred lift $A \to X_1$. Define $C_1$ to be the mapping cone of this map, so we have a cofiber sequence $A \to X_1 \to C_1$. We now iterate the forgoing to get a sequence of spaces
$$
\cdots \to X_n \to \cdots \to X_1 \to X_0 = X
$$
together with compatible maps $A \to X_n \to C_n$. Under mild hypotheses, the map
$$
A \to {\text{holim}}_n \phantom{,} X_n
$$
is a weak homotopy equivalence (for example, it will be the case when $A$ and $C$ are $1$-connected).

The homotopy groups tower above gives rise to a spectral sequence where the $E_1$-term is
given by $\pi_p(X_q,A)$ (I am being sloppy about the indexing conventions here). In a certain stable range (roughly, $p < 2(qa + c)$ if I recall, where $A$ is $a$-connected and $C$ is $c$-connected), these latter groups are identified with
$$
\pi_p(A^{[q]}\wedge \Omega C)
$$
where $A^{[q]}$ is the $q$-fold smash product of $A$ with itself.

A problem, which I'm not sure how to solve, is how does one identify the $k$-invariants of the tower to obtain a description of the $E_2$-term in this stable range?

*Added:* On second reflection, the above is not really an analogue of the Serre spectral sequence. The Serre spectral sequence for $F \to E \to B$ is given by filtering $B$ as a CW
complex and pulling back the fibration along skeleta. Dually, the thing to do would be to approximate the cofibration $A\to X \to C$ by a Postnikov tower $A^i$, and the take cobase change of $A \to X$ along $A\to A^{(i)}$ to get a tower $X^{(i)}$ with compatible maps from $X$ into it. Under mild hypotheses, this ought to converge to $X$ when taking the homotopy inverse limit. Then the thing to do would be to consider the homotopy spectral sequence of the tower $X^{(i)}$. However, I'm skeptical that there's a description of the $E_2$-term in terms of $A$ and $C$.