In a nutshell
The name Kleene closure is clearly intended to mean closure
under some string operation.
However, careful analysis (thanks to a critical comment by the OP
mallardz), shows that the Kleene star cannot be closure under concatenation, which
rather corresponds to the Kleene plus operator.
The Kleene star operator actually corresponds to a closure under the
power operation derived from concatenation.
The name Kleene star comes from the syntactic representation of the operation with a star *, while closure is what it does.
This is further explained below.
Recall that closure in general, and Kleene star in particular, is
an operation on sets, here on sets of strings, i.e. on languages. This will be used in the explanation.
Closure of a subset under an operation always defined
A set $C$ is closed under some $n$-ary operation $f$ iff $f$ is always
defined for any $n$-tuple of arguments in $C$ and
$C=\{f(c_1,\ldots,c_n)\mid \forall c_1,\ldots,c_n \in C\}$.
By extending $f$ to sets of values in the usual way, i.e.
$$f(S_1,\ldots,S_n)=\{f(s_1,\ldots,s_n)\mid \forall s_i\in S_i. 1\leq
i\leq n\}$$
we can rewrite the condition as a set equation:
$$C=f(C,\ldots,C)$$
For a domain (or set) $D$ with an operation $f$ that is always defined
on $D$, and a set $S\subset
D$, The closure of $S$ under $f$ is the smallest set $S_f$
containing $S$ that
satisfies the equation:
$S_f=\{f(s_1,\ldots,s_n)\mid \forall s_1,\ldots,s_n \in S_f\}$.
More tersely with a set equation, the closure of $S$ under $f$ may be defined by:
$$S_f \text{ is the smallest set such that } S\subset S_f \text{ and }
S_f=f(S_f,\ldots,S_f)$$
This is an example of least fixed-point definition, often used in
semantics, and also used in formal languages. A context-free grammar
can be seen as a system of languages equations (i.e. string set equations),
where the non-terminal stand for language variables. The least
fixed-point solution associate a language to each variable, and the
language thus associated to the intial symbol is the one defined by
the CF grammar.
Extending the concept
The closure as defined above is only intended to extend a subset $S$
into a minimal set $S_f$ such that the operation $f$ is always
defined.
As remarked by the OP mallardz, this is not a sufficient explanation,
since it will not include the empty word $\epsilon$ in $S_f$ when it
is not already in $S$. Indeed this closure corresponds to the
definition of the Kleene plus + and not to the Kleene star *.
Actually, the idea of closure can be extended, or considered in different ways.
Extension to other algebraic properties
On way to extend it (though it is no longer called closure)
considers more generally an extension to a set $S_f$ having specific
algebraic properties with respect to the operation $f$.
If you define $S_f$ as the smallest set containing $S$ that is a
Monoid for the binary function $f$, then you require both closure and a
neutral element which is the empty word $\epsilon$.
Extension through a derived operation
There is a second way which is more properly a closure issue.
When you define the closure of $S\subset D$, you can consider it
with respect to some of the arguments, while you allow values from
the whole set $D$ for the other arguments.
Considering (for simplicity) a binary function $f$ over $D$, you can
define $S_{f,1}$ as the smallest set containing $S$ that satisfies
the equation: $$S_{f,1}=\{f(s_1,s_2)\mid \forall s_1\in
S_{f,1}\wedge\forall s_2\in D\}$$
or with set equations:
$$S_{f,1} \text{ is the smallest set such that } S\subset S_{f,1} \text{ and }
S_{f,1}=f(S_{f,1},D)$$
This also makes sense when the arguments do not belong to the same
set. Then you may have closure with respect to some arguments in one
set, while considering all possible values for the other arguments
(many variations are possible).
Given a Monoid $(M,f,\epsilon)$ $-$ for example the monoid of strings
with concatenation $-$ where $f$ is an associative binary
operation on the elements of the set $M$ with an identity element
$\epsilon$, you can define the
powers of an element $u\in M$ as:
$$\forall u\in M.\;
u^0=\epsilon\; \text{ and }\; \forall n\in\mathbb N\; u^n=f(u,u^{n-1})$$
This exponentiation $u^n$ is an operation that takes as
argument an element of $M$ and a non-negative integer of $\mathbb
N_0$.
However, the natural extension of this operation to subsets of $M$ is
not the usual one which would be, for a given value of $n$,
$U^n=\{u^n\mid u\in U\}$. It should rather take into account the
original definition of $u^n$ from the operation $f$, wich would give:
$$\left\{
\begin{array}{l}
U^0=\{u^0\mid u\in U\}=\{\epsilon\}\\
\forall n\in\mathbb N,\; U^n=f(U,U^{n-1})
\end{array} \right.$$
so as to be consistent with the natural extension of the operation $f$
to subsets of $M$.
Now we can define the closure of $U_{\wedge,1}$ of $U\subset M$ for
the first argument of the power operation, as indicated above with
the set notation, as:
$$U_{\wedge,1} \text{ is the smallest set such
that } U\subset U_{\wedge,1} \text{ and }
U_{\wedge,1}=f(U_{\wedge,1},\mathbb N_0)$$
And this does give us the the Kleene star operation when the
construction is applied to the concatenation operation of
the free Monoid of strings.
To be completely honest, I am not sure I have not been cheating. But
a definition is only what you make it, and that was the only way I
found to actually turn the Kleene star into a closure. I may be
trying too hard.
Comments are welcome.
Closing a set under an operation that is not always defined
This is a slightly different view and use of the concept of closure. This view is not really answering the question, but it seems good to keep it in mind to avoid some possible confusions.
The above implies that the function $f$ is always defined in the
reference set $D$. That may not always be the case. Then closure can
also be a mathematical technique to extend a set so than some
operation will always be defined. The way it works in practice is as
follow:
start with the set $D$ where $f$ is not always defined;
build another set $D'$ constructed from elements of $D$, with an
operation $f'$ that is always defined, such that you can ...
show that there is an isomorphism between $D$ and a subset of $D'$
that is such that $f$ is the image of $f'$ restricted to that
subset.
Then the set $D'$ with the operation $f'$ is a closed extension of $D$
with $f$.
That is how integers are built from natural numbers, considering the
set of pairs of natural numbers quotiented by an equivalence relation
(two pairs are equivalent iff the two elements are in the same order
and have the same difference).
This is also how rationals can be built from the integers.
And this is how classical reals can be built from the rationals,
though the construction is more complex.
