Currently, the only primary weapons in the game that do special damage (Solar, Arc, and Void) are Raid weapons; Atheon's Epilogue, Praedyth's Timepiece, Fatebringer, and that scout rifle (I can't remember the name).
I have to wonder; why is it that Legendary guns get special damage, when Exotics don't?
Here's what I'm thinking the should do for some of the Exotic primaries:
[b]Red Death[/b] + Solar Damage
[b]Hard Light[/b] + Arc Damage
[b]Thorn[/b] + Void Damage*
*Thorn would still fire the same green tracers as always, but Void damage would be applied to them; it should also be applied to the DoT effect you get from the special perk.
There's also The Last Word, SUROS Regime, and Bad Juju, but I'm not sure what they should get. You let me know.
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Suros Regime actually USED to have solar elemental damage on it, back in the early release, along with most(not all, I think) of the other exotic primaries. Then Bungie took it away because they wanted the raid weapons to be "unique". Which they still WOULD be, since they would be the only LEGENDARY tier primaries that do elemental. Just another bone-headed decision by the dev team in charge of Destiny.