The best replacement for openid is oauth.

The best alternatives to openid are rated as following:
★★★★★  oauth
★★★★  saml
★★★  ldap
★★★  facebook connect
★★★  shibboleth
★★★  cas
★★  openauth
★★  cardspace
★★  opensocial
★  oauth 2
★  openid connect
★  sso
★  persona
★  kerberos
  openam
  browserid
  mozilla persona

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No companions for openid have been suggested so far.
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Google Trends

Scale is based on the average worldwide traffic of openid in all years. Click on graph for more.
openid
oauth
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