I don't think there is a direct way by which you can identify the name of the list. But, generally the list would be related to the module in the dashboard and you can identify the list from the dimensions in the module.
As @Yash1 mentioned, there is no direct way to identify which list has been published to dashboard.
But generally, we publish lists to dashboard as page selectors to sync the module from the drop down. "Synchronize selection" option works when the page selectors in the dashboard and the dimensions in the modules are same list.
So if you provide access to all the lists which are used in the modules published to the dashboard, it should be good.
I just want to first clarify that we don't give users access to lists corresponding to page selectors published to the dashboard (Role ==> Lists) but rather to the module from which they were published (Role ==> Modules)
Like others said, currently there is no way to confirm with 100% certainty the module from which the page selector was published. Ideally, you should publish the page selector from one of the modules on the dashboard, however, there is nothing to prevent a modular from publishing a page selector from a module that is not published on the dashboard ( I don't recommend it but I've seen it when debugging the issue with others)
There are 2 ways you can resolve the issue of determining the module of the page selectors published:
From the Module settings page, look at the applies to column to find the modules that have the list of the page selector in question. that will help you narrow the modules that could have been used to publish the page selector.
Delete the page selector and republish it to the dashboard from a module on the dashboard. (my preferred method)
Maybe this is something worth recommending to the product team.
Einas "Give a Man a Fish, and You Feed Him for a Day. Teach a Man To Fish, and You Feed Him for a Lifetime"
According to the Planual, you should give users access to lists in very limited situations. For example; if you have a numbered list where you allow the user to add items to the list (New employee, product,...etc) then, in this case, you should give the user role write access to the list. When you think about it, Changing lists is not something the end-user will - or should- do.
How can I publish a specific page selector from a sys module. For example - I have a sys country module, but when I choose to publish as page selector it publishes along with the country and not just region. Maybe that was the reason why I published from the region list and not sys module.