One of the things I've always wanted to see happen in Destiny was a bigger focus on the faction rivalries. There is some interesting lore behind each group, but hardly any of that shows up in game. I think SRL could provide an opportunity to give the factions a little flavor. It wouldn't be enough for the lore nuts, but I think it could be a cool way for Bungie to incorporate the existing lore into a fun, new game mode.
My idea would be to give people racing gear with faction logos and colors (please mix up the color schemes though, so not everyone from a given faction looks exactly alike). Also, if SRL becomes an "event", Bungie could do a faction competition based on how people perform. I'm not sure how that would all work out, and balancing out each faction would be a huge challenge, but I think it would be cool.