I had tried booting a 5.1 install DVD back in August 2012, with my existing ESXi 5.0 already installed the thumb drive. But the upgrade it attempted failed a few minutes into it, with an error I don't recall. So I simply tried again, and chose install this time instead of upgrade, and it worked, overwriting 5.0 with 5.1, all set. I wanted to try a fresh start anyway, to see how well the LSI baked in drivers worked, keeping this simple.
But you may wish to upgrade, particularly if you've made changes to your ESXi 5.0 configuration that are hard to re-do.
If an easy upgrade approach appeals to you, this brief article points you toward two different ways of doing this, both with the same end result. Both of these methods are very similar to the technique I've outlined in various LSI RAID adapter articles on this site.
These methods should work even for a licensed install of ESXi 5.0, but I haven't verified that.
If you wind up trying either of these methods below, help everybody else out by briefly letting us know how it went for you. Just drop us a comment below, no sign-in required!
ESXi Free – how to upgrade to ESXi 5.1, by Vladen Seget, 2012
download the big zip file, SSH it into ESXi 5.0, type a long esxcli command, reboot
A Pretty Cool Method of Upgrading to ESXi 5.1, by William, September 18 2012
even simpler, esxcli command pulls the code down then applies it, reboot