TITLE: Unique existence and the axiom of choice
QUESTION [28 upvotes]: The axiom of choice states that arbitrary products of nonempty sets are nonempty. 
Clearly, we only need the axiom of choice to show the non-emptiness of the product if 
there are infinitely many choice functions. If we use a choice function to construct a mathematical object, the object will often depend on the specific choice function being used. So constructions that require the axiom of choice often do not provide the existence of a unique object with certain properties. In some cases they do, however. The existence of a cardinal number for every set (ordinal that can be mapped bijectively onto the set) is such an example.

What are natural examples outside of
  set theory where the existence of a
  unique mathematical object with
  certain properties can only be proven
  with the axiom of choice and where the
  uniqueness itself can be proven in ZFC (I
  don't want the uniqueness to depend on
  a specific model of ZFC)? 

The next question is a bit more vague, but I would be interested in some kind of birds-eye view on the issue.

Are there some general guidelines to understand in which cases the axiom of choice can be  used to construct a provably unique object with certain properties?

This question is motivated by a discussion of uniqueness-properties of certain measure theoretic constructions in mathematical economics that make heavy use of non-standard analysis. 
Edit: Examples so far can be classified in three categories: 
Cardinal Invariants: One uses the axiom of choice to construct a representation by some ordinal. Since ordinals are canonically well ordered, this gives us a unique, definable object with the wanted properties. Example: One takes the dimension (as a cardianl) of a vector space and constructs the vector space as functions on finite subsets of the cardinal (François G. Dorais).
AC Properties: One constructs the object canonically "by hand" and then uses the axiom of choice to show that it has a certain property. Trivial example:  $2^\mathbb{R}$ as the family of well-orderable sets of reals.
Employing all choice functions: Here one gets uniqueness by requiring the object to contain in some sense all objects of a certain kind that can be obtained by AC. Examples: The Stone-Čech compactification as the set of all ultrafilters on it (Juris Steprans), or the dual space of a vector space, the space of all linear functionals. (Martin Brandenburg) The AC is used to show that these spaces are rich enough. Formally, these examples might be categorized in the second category, but they seem to have a different flavor.

REPLY [2 votes]: This may be a bit trivial an answer to this (old) question, but since a number of people have reacted with surprise when I mentioned this, and it seems to fall within the purview of what is being asked, let me state:
The Axiom of Choice is equivalent to the following statement:

If $X$ is a set, $\sim$ an equivalence relation on $X$, and $I$ a set, then the obvious map $\Phi\colon X^I/(\sim^I) \to (X/{\sim})^I$ is a bijection.
(Here $X^I$ stands for the set of all $I$-indexed families of elements of $X$ and $\sim^I$ is the equivalence relation which holds between $(x_i)_{i\in I}$ and $(y_i)_{i\in I}$ in $X^I$ iff $x_i \sim y_i$ for all $i\in I$; and $\Phi$ takes the class of $(x_i)_{i\in I}$ to $([x_i])_{i\in I}$, where $[x]$ denotes the class of $x$ mod $\sim$.)

Note that, independently of AC, $\Phi$ is injective by definition of $\sim^I$.
(The equivalence with AC of the above is almost obvious.  If AC holds, then we can lift $(\bar x)_{i\in I}$ in $(X/{\sim})^I$ to a family $(x_i)_{\in I}$ in $X^I$ by choosing a representative for each class.  Conversely, if the above holds and $(Z_i)_{i\in I}$ is a family of inhabited sets, let $X$ be their disjoint union, $\sim$ the equivalence relation for which $Z_i$ is the partition of $X$ so that $X/{\sim}$ is $I$, and use the above to lift the “identity” element of $(X/{\sim})^I = I^I$ to an element of $X^I$ to see that $\prod_{i\in I} Z_i$ is inhabited.)
Thus, in the above context, AC tells us that an injective map $\Phi$ is surjective, or, if we want, that every element in the target has a (necessarily unique) antecedent by $\Phi$.