If you are within your 14 day return period you can - assuming there is a larger capacity model available. Currently the iPhone 5S comes in the 16 and 32 GB varieties. Outside of the 14 day return period, no, you cannot. And there is no way to expand the storage of an iPhone.
Did you try a cloud-based storage option, like Dropbox, etc? A free account with most of the cloud providers will give you a couple of GB to work with, and for a few dollars a month you can get enough space to keep your photos and media there. The only caveat is that to use these files you are also using data.... sometimes lots of it.
I use the Mophie Space. It isn't the same thing as having more internal storage but it does help a lot. I keep all music and photos on the Mophie. That frees up a lot of space on the 5s