Why do we need a $G$-universe? Let $G$ be a compact Lie group. Before defining $G$-prespectra, we have to define a $G$-universe $\mathcal U$.
Question: Why do we need a $G$-universe?
A $G$-universe is defined to be a countably infinite-dimensional (real) representation of $G$ with an inner product such that


*

*$\mathcal U$ contains the trivial representation.

*$\mathcal U$ contains countably many copies of each finite-dimensional subrepresentation.


After fixing a $G$-universe, we can define a $G$-prespectrum (indexed on $\mathcal U$): a collection $\lbrace EV \rbrace_V$ of $G$-spaces indexed by finite-dimensional subrepresentations $V$ of $\mathcal U$ together with $G$-maps $\sigma_{V,W}:\Sigma^{W-V}EV \to EW$ for pairs $V \subset W$. 
(For details and relevant papers, see "Basic notions in equivariant stable homotopy theory".)
A $G$-universe seems to be used only for taking (co)limits and to be regarded as a source of finite-dimensional (orthogonal) representations of $G$. I feel that it is very artificial to use a $G$-universe for the purposes. 
For example, what is wrong if we deal with the category of (equivalent classes of) finite dimensional (orthogonal) representations of $G$? This is a small category, so we do not have any problem to consider (co)limits.
 A: The short answer is, I think, that one doesn't really need a $G$-universe, and that one could also index a $G$-spectrum over all finite dimensional representations of $G$.
A: Tyler, you are too fast: didn't give me a chance to answer first!
Of course, I agree with everything you say.  I wrote the following
before seeing your answer (except for the last paragraph).
Since I introduced this choice, let me explain.  But first, echoing
André, taking equivalence classes would be a wrong choice
even if it gave a category, which it doesn't.   In fact, the word 
“representation” in this context is a convenient lie:
we are not doing representation theory here, and we must not think
at all in terms of equivalence classes.  For  example, isomorphisms
between “representations” control signs in equivariant
cohomology theory. 
One point is to obviate set theoretic nonsense.  It has become 
unfashionable, perhaps, to pay attention to this, but of course
the collection of all finite dimensional representations is not
a set, and for many purposes, such as taking colimits as you
say, one does want a set.
A mathematical point is that different universes give different
categories of G-spectra, and that matters enormously: change of
universe plays an essential role in equivariant stable homotopy
theory.  This could be dealt with in other ways, but use of 
universes is convenient. 
Actually, how essential a universe is, depends on which choice 
of a category of $G$-spectra one has in mind.  For all choices,
it is very convenient to work with $G$-vector spaces with a 
fixed given $G$-inner product.  For orthogonal $G$-spectra, the fact 
that the category $\mathcal I$ of such $G$-inner product spaces 
is essentially small (equivalent to a small category) allows us to
use it without actually specifying a universe, although one does
obtain a different $\mathcal I$ for each choice of a set of irreducible
representations (the complete universe, allowing all, being the
most important).
For $G$-spectra in the sense of Gaunce Lewis and myself, and therefore
for the $S$-modules of EKMM (Elmendorf–Kriz–Mandell–May) use of a 
universe is truly essential: $G$-spectra are obtained from $G$-prespectra
as colimits over inclusions of sub $G$-inner product spaces of a universe.
Such colimits make no sense without use of some device to ensure smallness.
In this line of development, use of a universe seems truly essential.  The
linear isometries $G$-operad $\mathcal L$ is central to the construction of the 
smash product (and to lots of work in equivariant infinite loop space 
theory), and $\mathcal L(j)$ is the $G$-space of linear isometries $U^j\to U$,
where $U$ is the universe in which one is working.  It would be ludicrous 
to try to make sense of that without working in a universe.
As a philosophical point, it is essential to be eclectic in this area and to
allow use of different categories of $G$-spectra, such as orthogonal and Lewis–May 
or EKMM, since there are many things that one can readily prove with one and not 
the other.  For a comparison of these two and discussion of change of universe, see 
for example Mandell–May, Equivariant orthogonal spectra and $S$-modules.
That source explains how, in orthogonal $G$-spectra, one can actually work with one 
fixed universe, even the trivial one, and obtain equivalent categories as Tyler
says.  Hill–Hopkins–Ravenel took that observation from Mandell–May and ran with it.
To be honest, I sometimes regret we made that observation; as Tyler notes, it can
be a source of confusion, and it can sometimes obscure the mathematics.  Here again
it is wise to be eclectic and think in terms of both physical change of universe
and “phantom” change of universe in terms of that observation.
A: I suspect that Prof. May will have the definitive answer here.  However, here are some of the more mundane problems that the choice of universe addresses.


*

*It cuts down on set-theoretic issues, by describing an equivariant spectrum as a functor on a small category.

*Back when Lewis-May-Steinberger put equivariant stable theory on a firm footing, one should note that prespectra and spectra were not functorial in representations: we only had structure relating $EV$ and $EW$ when $V \subset W$.  The functoriality in isomorphisms only came later.

*Getting a little confused about this last point was the source of some mistakes, and if you want to read a scathing criticism of this (such as people casually writing $S^{V-W}$ when $V$ and $W$ are unrelated $G$-representations) then you can look at Adams' "Prerequisites for Carlsson's lecture", section 6.

*It helps to understand the smash product.  The smash product of a spectrum $E$ indexed on ${\cal U}$ and a spectrum $F$ indexed on ${\cal U'}$ is most naturally a spectrum $E \overline\wedge F$ indexed on ${\cal U \oplus U'}$, because it is built out of $EV \wedge FV'$ for $V \subset {\cal U}$ and $V' \subset \cal U'$. This exterior wedge is nicely associative and commutative, and so this concentrates the "issues" with the smash product into the problem of internalizing the smash power.

*It conveniently describes which type of equivariant spectrum you have by describing which $G$-representations you are allowing formal deloopings by.  Different isomorphism classes of universe lead to quite different theories.

*Finally: Just as André says, you don't technically need a universe anymore.  "Orthogonal spectra" are essentially some kind of functor defined on representations, and you may find this version palatable.  In fact, the Hill-Hopkins-Ravenel construction of the norm critically used that $G$-equivariant orthogonal spectra form a category equivalent to a category of plain-old ordinary spectra equipped with a $G$-action, and so representation spheres didn't even appear.  (I can't speak for anyone else, but this caught me by surprise.  My confusion turned out to be that you have several categories that are equivalent and which look basically the same, but have quite different homotopy categories and homotopy theories because these equivalences of categories do not respect the notion of "weak equivalence" at all.)
