I would like to add a few points:

1. You can define $RO(G)$-graded homotopy groups of $G$-spectra, see for example [Stefan Schwede's course notes on equivariant homotopy theory.][1] 

2. These groups are interesting. For example the inclusion of the 0-skeleton $S^0\rightarrow S^{\sigma}$ of the sign representation stabilizes to a non-trivial (and non-nilpotent!) element in $\pi_0^{C_2} S^{\sigma}\cong \pi_{-\sigma}^{C_2} S^0.$  This is proved by showing its Hurewicz image in Bredon homology is non-trivial.
3. At least for finite groups $G$-representation spheres can be triangulated and constructed out of the standard $G$-cells ($G/H_+\wedge S^n,$ $G/H_+\wedge D^n$).  So ordinary equivariant weak equivalences between cell complexes induce $RO(G)$-graded equivalences.  

4. One reason to put representation spheres into your theory is so that you have a reasonable form of equivariant Spanier-Whitehead duality.  At the very least we want finite G-sets to be dualizable, so we want them to equivariantly embed into a G-sphere, which should be one of the spaces in the sphere spectrum.  This will not work if we only allow trivial spheres.  

  [1]: http://www.math.uni-bonn.de/~schwede/equivariant.pdf%20%22Stefan%20Schwede's%20lectures%20on%20equivariant%20homotopy%20theory%221.1.