Hi Keith,
Supposed that you have at least one public IP address on any of the interfaces, you may use any public DDNS service (Dynamic DNS), to register a name bound to the interface, and thus keep current name to IP resolution. Then use the name you have registered for the 'Host' parameter in the client's connection configuration.
Here is a link to the IOS configuration guide, feature has been available since 12.3(14)T: http://www.cisco.com/en/US/partner/docs/ios/ipaddr/configuration/guide/iad_ddns_ps6441_TSD_Products_Configuration_Guide_Chapter.html
HTH
Regards,
Vasil
