Our vCenter 5.1 evolution expired, I'm look solution to extend it without buy license.
If you need to have more time for your evaluation you could contact you local VMware Sales team and kindly ask if they can give you an extension.
If not then the only way is to install a valid license since the vCenter license is not for free.
Can I re-install the vCenter to enter new evolution key?
A fresh installation (either on a new host or after deleting all traces of the old installation) may work to start the evaluation period of 60 days again. However it's not only vCenter Server which expires it's also the ESXi hosts which you may need to reinstall.
Btw. there's no license key required for evaluation.