As an originals artist, I sometimes shy away from doing too many covers. But while I enjoy performing mostly my songs in my set, sometimes there are gigs that call for a more covers-oriented set list, like if I'm playing at a restaurant or happy hour. In the case of those type of gigs, I will play either 50% covers 50% originals, or 75% covers 25% originals, depending on how long they want me to play. Those gigs usually pay better than originals-only gigs, so I've started to not shy away from them, though I want to get back into writing new material now that I've learned a few hours worth of covers to add to my repertoire.
When choosing covers to add to my set list, I have two rules: 1) pick "unlikely" covers that aren't often covered by everyone else and 2) pick from some of my favorite songs/acts. Some of my favorite songs that I've chosen to cover include: Fade Into You (Mazzy Star), Ring of Fire (Johnny Cash) and You're So Vain (Carly Simon). Some of the covers from the more "unlikely" category include Did I Shave My Legs For This (Deanna Carter), Fake Plastic Trees (Radiohead), and Strange Condition (Pete Yorn). And of course I sprinkle in some well known classic rock or country stuff that people will often sing along to, like Bad Moon Rising (CCR) and Take It On The Run (REO Speedwagon).
As I'm still predominantly an "originals act", I don't want to go too far into learning hundreds of covers, and then feel I'm just a human jukebox. My goal is always to present something new or different that other people aren't doing - whether it's my songs or other people's songs. To me that method satisfies me as both an artist and makes it feel less like work, and the most like fun.