This turned out to be much more manual work than I hoped: renaming a VM and the VMDK used by the VM in ESXi.
If you prefer to do all steps by hand on the command-line plus two vi sessions for the vmx* files and the main vmdk file: Just one more esxi-guy: How to Rename a Virtual Machine with the CLI in esxi 5.0.
If you have vCenter, you can do it from the guy. Note for Free ESXi users: vCenter is a payed product, but there is a trial of vCenter.
A bit less manual work, but a bit more scripting from the console (if you are comfortable with that) is at Howto: Rename a VM – Yellow Bricks to which you might want to add the script mentioned at Rename multiple files by replacing a particular pattern in the filenames using a shell script – Stack Overflow.
You can also script from vMA or vCLI: VMware KB: Renaming a virtual machine disk (VMDK) via the vSphere Management Assistant (vMA) or vSphere CLI (vCLI).
–jeroen