What do we call the covector associated to a vector? Let $V$ denote an inner product space. Write $V^*$ for either the algebraic dual, or else the continuous dual. In either case, for each vector $v \in V$, we get a covector $v^c \in V^*$ given by:
$$v^c = \langle v,-\rangle$$

Question. We call $v^c$ the [what] of $v$?

 A: Echoing @Lærne, $v^c$ is the dual of $v$.  If you want a reference, how about Halmos's "Finite-Dimensional Vector Spaces"?  In the second edition, sections 13, 14, and 15 along with sections 67, 68, and 69 may be what you're looking for.
A: $\renewcommand{ket}[1]{|#1\rangle}$
$\renewcommand{bra}[1]{\langle#1|}$
There are several common terminologies:


*

*As you might guess based on the fact that $V^*$ is called the "dual" of $V$, in mathematics and mathematical physics the covector of a vector $v$ is called the "dual" of $v$.

*In physics, vectors are often denoted e.g. $\ket{v}$.
The $v$ is a label which names the vector.
In other words, $\ket{v}$ is like $\vec{v}$.
The associated covector is denoted $\bra{v}$.
The notation is justified by the fact that if you line up the covector and the vector you get
$$ \bra{v}v\rangle$$
which is the usual symbol for an inner product.
The symbol $\ket{\cdot}$ is called a "ket" and the $\bra{\cdot}$ is called a "bra" so that when you line them up to make an inner product you get "braket".
In this system, the association between the vector and associated covector is expressed by using the same label, $v$, in either the ket or the bra symbol.
In speech we just call $\bra{v}$ "bra $v$", as opposed to "ket $v$".
I think many people would still say "$\bra{v}$ is the dual of $\ket{v}$".

*You mentioned inner product spaces.
In fact, the inner product function $\bra \cdot \cdot \rangle$ can be thought of as a rank-2 tensor: when you feed it two vectors it spits out a scalar.
This tensor can be denoted in index notation as $g^{\alpha \beta}$.
The inner product of two vectors $U_\alpha$ and $V_\beta$ is written
$$\bra{U}V\rangle = g^{\alpha\beta} U_\alpha V_\beta \tag{1}$$
where matching indices indicates contraction.
The upper indices are called "covector" indices because they indicate the need to consume a vector in order to produce a scalar.$^{[a]}$
So, if we have something like this
$$U^\alpha \, ,$$
that thing must be a covector because it contracts with a single vector to yield a scalar.
Therefore, upper indices indicate covector parts of the tensor and lower indices indicate vector parts.
Now, if we contract $g$ with only one vector
$$ g^{\alpha \beta}V_\beta$$
we're left with a free upper index, i.e. a covector.
This thing is usually denoted $V^\alpha$ and is precisely the covector associated to $V_\beta$, a.k.a the dual of $V_\beta$.
In this system you might also call $V^\alpha$ the "covariant version of $V_\beta$" or say that $V^\alpha$ is "$V_\beta$ with its index raised".
The point of this answer is to list the various common ways that the covector versions of vectors are called, and provide some motivation for the terminology.
$[a]$: It's also common to call upper indices "covariant" and lower indices "contravariant". This has to do with how the components of representations of vectors and covectors, expressed in a given basis, transform under coordinate transformations.
