I don't know about a drop down list but you could make the Options be the names of who they could select to cover the shift and then below that have condtions that say 'SelectedOption - is equal to - Name1' then update the list with that name, etc.
So on your "Send email with options" part the User Options would be "Name1, Name2, Name3" and so on.
I have one similar to this but I did not go and try it for your scenerio but it should work.
Another way of approaching this would be to have a SharePoint list as the point of original entry. This list has a person or group column (required) where the person covering the shift is specified.
Here is how this might work:
Will this solution meet your needs?
The only other thing I have done (that I consider a work around) is I have a flow that goes out as an approval instead of the email with options.
For mine it goes out to everyone who can pick up an item. If they don't pick it up they do nothing and if they do they click Accept.
Next it looks for a Condtion of "Accept" and if they do then it sends out an email to everyone on the list letting them know that Name1 picked up this item.
Then it updates the item on my SharePoint List. It does two things there: 1) changes the item status to "Assigned" and 2) it stores the "Approver name" (which is collected from the Approval email that is sent out) as a Text field on my sharepoint list (It will not store it as a people field...or I didn't get it to).
Not sure if that will work for your needs but it does work on what I am using it for and is similar to yours.
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