"Logically equivalent formulae express the same _______." <- What word do logicians use for the blank? Meaning denotes the truth conditions of a sentence: what would have to be the case for the interpreted formula to be true. Nevertheless, without an interpretation, two logically equivalent formulae express something that makes them equivalent. What is the name for that?
 A: I think that this one is a "slippery" question.
We can approach it from different points of view, with increasing degree of "difficulty" and, so to say, "philosophical committment".
First approach : is it possible to avoid to be "involved with" meaning in the elucidation ?
Stay, for simplicity, with propositional logic; we have that :

$\varphi \leftrightarrow \psi$ iff $\vdash \varphi \rightarrow \psi$ and $\vdash \psi \rightarrow \varphi$.

If we assume modus ponens, this amount to "interderivability", i.e. :

$\varphi \vdash \psi$ and $\psi \vdash  \varphi$.

There is no "smell" of meaning here. But, without meaning, i.e. without interpretation of the above formulae, what is "expressed" by their equivalence ?
Second approach : there is a long tradition in modern logic regarding "extensionality". From Frege's Begriffsschrift onwards, mathematical logic has been concerned mainly with truth-functional contexts.
According to Frege, the Bedeutung of an entire proposition is its truth-value, either the True or the False. For Frge, complete propositions, like names, have objects as their Bedeutungen, and in particular, the truth-values the True or the False. In this way, he is able to transcribe sentential connectives such as “and” and “or,” etc., as truth functions in the strictest sense — functions that take truth-values as argument and yield truth-values as value. [see Kevin Klement, Frege and the Logic of Sense and Reference (2002)]
According to W&R [see Alfred North Whitehead & Bertrand Russell, Principia Mathematica to 56 (2nd ed - 1927), page 115] :

It is obvious that two propositions are equivalent when, and only when, both are true or both are false. Following Frege, we shall call the truthvalue of a proposition truth if it is true, and falsehood if it is false. Thus two propositions are equivalent when they have the same truth-value.

AIt should be observed that, if $p \equiv q$, $q$ may be substituted for $p$ without altering the truth-value of any function of $p$ which involves no primitive ideas except those enumerated in *1 [the truth-functional connectives].

We shall give the name of a truth-function to a function $f(p)$ whose argument is a proposition, and whose truth-value depends only upon the truth-value of its argument. All the functions of propositions with which we shall be specially concerned will be truth-functions, i.e. we shall have

$p \equiv q . \supset .f(p) \equiv f(q)$.

The reason of this is, that the functions of propositions with which we deal are all built up by means of the primitive ideas of *1. But it is not a universal characteristic of functions of propositions to be truth-functions. For example, "A believes $p$" may be true for one true value of $p$ and false for another.

In "modern" terms, this is the so-called Replacement theorem [see S.C.Kleene, Introduction to Metamathematics (1952), page 116] :

If $A$ and $B$ are formulas, $C_A$ is a formula constructed from a specified occurnce of $A$ using only the [truth-functional connectives], and $C_B$ results from $C_A$ by replacing this occurrence of $A$ by $B$, then $A \leftrightarrow B \vdash C_A \leftrightarrow C_B$.

And equivalently : $A \leftrightarrow B, C_A \vdash C_B$.
In a "universe" of "extensional" contexts, that considered by the  "simplified" semantics of logical languages, it is easy to "equate" the condition for two equivalent formulae of having the same "truth conditions" with that of expressing "the same thing".
It is likely then, that "that thing" is their meaning.
Third approach : see Michael Dummett and his The Logical Basis of Metaphysics (Harvard UP, 1991) and Proof-Theoretic Semantics.
The basic idea - I think - is to build a theory of meaning without truth. But I'm not prepared to discuss it.
A: I would still go for meaning here: The meaning of a formula is a function that takes a structure and a valuation for the formula's free variables, and produces a truth value.
Logically equivalent formulas have the same meaning in this sense.
A: Logicians use the word semantic to refer to the "meaning" of a formula. It is the interpretation of the formula in a given structure.
For instance take the First-Order language with $+,*,0,Succ$ (the classic Peano language). For instance $\varphi(x)=(\exists y. y*y=x)$ is a formula in this language.
If we interpret $\varphi$ in the structure $(\mathbb N,+,*,0,Succ)$, then the semantic of $\varphi$ is the set of square numbers (the set of $x$ that makes $\varphi(x)$ true). But we can also interpret $\varphi$ in $(\{0,1\},\vee,\wedge,0,Succ)$ in which case its semantic is the set $\{0,1\}$. If a formula $\psi$ does not contain free variables, then its semantic is imply $true$ or $false$, and still can depend of the structure For instance the formula $\exists x,y,z: x\neq y\wedge x\neq z\wedge y\neq z$ states that the structure has at least $3$ elements, so it is true on $\mathbb N$ but false on $\{0,1\}$.
So the semantic of $\varphi$ depends on the structure on which it is interpreted. Two formulae are logically equivalent if they have the same semantic under all structures. It means that their equivalence is not an accident due to the chosen structure, but is "intrinsic" to the formulae: they express the same thing, regardless of what is the precise meaning of the operators.
A: Usually, two logically equivalent formulas express interderivability, or one could say that the formulas are interderivable.  In other words, in a given calculus with axioms, if A is logically equivalent to B, then if A gets assumed, then we can derive B using only the rules of inference and axioms of the calculus, and if we assume B, then we can derive A using only the rules of inference and axioms of the calculus.
So one might say that logically equivalent formulae express the same derivability.
For instance, in Lukasiewicz three-valued logic (p$\rightarrow$q) is equivalent with ($\lnot$q$\rightarrow$$\lnot$p) implying that if (p$\rightarrow$q) is interderivable with ($\lnot$q$\rightarrow$$\lnot$p) (note that this does NOT imply that [(p$\rightarrow$q)$\rightarrow$($\lnot$q$\rightarrow$$\lnot$p)] is a theorem NOR that [($\lnot$q$\rightarrow$$\lnot$p)$\rightarrow$(p$\rightarrow$q)] is a theorem here (though in fact they are theorems here), since Lukasiewicz three-valued logic does not have a deduction theorem which allows for reasoning to those theorems in this way).
Interderivability in some ways comes as a more general notion than logical equivalence, since logical equivalence only applies to well-formed formulas.  In contradistinction, interderivability applies to sets of well-formed formulas under a fixed rule.  For example, the axiom set {CpCqp, CCpCqrCCpqCpr, CCNpNqCqp} under detachment and uniform substitution is interderivable with the axiom set {CCpqCCqrCpr, CpCNpq, CCNppp} under detachment and substitution.  But, the set {CpCqp, CCpCqrCCpqCpr, CCNpNqCqp} under detachment and substitution is not equivalent to {CCpqCCqrCpr, CpCNpq, CCNppp} under detachment and substitution since neither set with its rules is a well-formed formula.
