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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Last word gets solar, suros gets arc, uni remote should deal a random element with every shot just for spits and giggles, bad juju should deal void, nlb should deal solar. I think that's all of them that weren't listed.