Work with what you’ve got. Play to your strengths (literally). If you have a great range, showcase it with big sweeping melodies (Queen, Mariah Carey, and other non-cheesy artists as well!). If your tone is average but you have a great sense of groove, cover more rhythmic songs that highlight that. Choose a song that could have been written for you.
Find your key. Just because the original singer can hit that low G doesn’t mean that’s what is best for you. Find your best range and make sure you are hitting your sweet spots. There are plenty of apps that can transpose your favorite song to your perfect key if you don’t want to do it yourself (or, there are people like me that can help you).
Do it your way. Unless you are playing a wedding and it’s the couple’s special song, don’t feel married to the original version. Try different embellishments. Find your voice and treat the tune like you wrote it.
Experiment with arrangement. Try changing the time signature/feel. (Here is my rendition of Outkast’s “Hey Ya” as a waltz, for example.) Make a rock song into a ballad, or vice versa. Put jazz chords to a simple pop song. Turn a heavily produced number into a minimalist piece (this is also a good way to go if you are still struggling with self-accompaniment).
Get permission. If you’re playing an open mic, this isn’t such a big issue. But if you plan on recording a cover and sending it out into the world, check out the info on licensing at the Harry Fox Agency.