I apologize for overlooking the fact that your site is using OneCommunity. That changes things quite a bit, since your OC Server manages how the user logs in, which includes their password/authentication model.
As you mentioned, Gigya has their own password reset process, which is what would need to be used by the member in order to re-access their account. If, for whatever reason, the member is unable to use that process you would need to work with Gigya to determine how to grant that user access to that account again. Unfortunately, that is entirely outside of our control as I am sure you understand.
Regarding the old Eve data, and how Eve was able to resend passwords, Eve used a two way encryption on member passwords, so that they were stored securely, but we were able to get the original value to resend in emails. When we exported the member data, we got the original passwords, and SHA-256 hashed them when providing them to Gigya. Those SHA-256 values are one way hashed and we no longer have the original Eve data, so there is no way for us to get what that members original password was. Again, this all falls back to Gigya, since your login integration is completely handled by them now, as part of OneCommunity.
Finally, the "Resend Registration Details" action is listed in Manage Members because when a member joins your community through OneCommunity, we give the site the opportunity to provide useful information to the member through that welcome email. A lot of communities will include directions in their welcome message about where to start on the site, or some background on everything. Sometimes users will need/want to get a new copy of that email, which is why that action still applies to your site even though you are using OneCommunity.
I hope this information helps, and I hope that you and Gigya are able to get this user back into their account.