I'm not sure entirely how to go about this.  A friend of mine asked me to install Dolphin for him a long time ago.  Recently, he'd been complaining about the site being slow and old.  I don't moderate his website but as a favor to him, I looked over his administrator dashboard.  Clearly, I knew that the script was outdated and wanted to go about upgrading to the latest script without wiping out all of his data.  Is this possible and how do I do it?  All the upgrade packages are for 7.x to later versions of 7.x.  What about from 6.1.4 to 7.x?

You will have to manually upgrade it from 6.1.4 to 6.1.5 and then to 6.1.6. There is a stock module in 7.X.X called "Data migration from Dolphin 6.1.6". I have never tried it but it's supposed to let you go from 6.1.6 to 7.