At Admin Panel -> Settings -> Switches, the "allow claiming topics via edit/register" switch enables a system whereby any guest or member can ask to claim any unowned topic to their account. On the details page of unowned topics there'll be an "Is this your topic? Update and claim it!" link at the bottom which leads to the edit topic page. There, the person -- whether an unregistered guest or a member -- will be asked to write a message with evidence that they own the site, such as an email address on the topic's website domain that you can contact them at. They can then make any changes to the topic. If they're an unregistered guest, they'll be asked to choose a username and password and the username, password and email fields will all be marked as required for them. When they submit their changes, it'll be held as a pending topic edit.
As administrator, you'll review claims at Admin Panel -> Validation under the Topic Edits tab. If you're requiring validation of topic edits in other situations, you'll notice the ones that involve a claim of ownership have a special warning message that includes a link you can use to send them an email asking for clarification or further evidence if needed. Once you approve the pending edit, the claimant becomes the owner and can find their topic in their member home area.
Note that member registration must be enabled for this to work.