Are you referring to an "Alternate Name" that is "Duplicated"?
Or, are you referring to a "Duplicate" individual/person in "Family Tree"?
There is quite a significate difference.
By the way, you most likely CANNOT "Delete" individual/person; ONLY, "Merge".
If you have TRUE duplicate names (i.e., two different PID records that contain similar vitals, parents, siblings, spouses, children, or residences indicating that the two PID records are representing the same exact person), you MERGE them together. This makes sense if they are representing the same person.
"Duplicate Names" is not an adequate justification by itself for merging records though. Close examination to both records to ensure that they are really representing the same person is essential before performing a merge.
Also note that even though the term "Delete" is used on the FamilySearch website when referring to the removal of a PID record from the general view of the database, the term is very misleading. Once a PID record has been created, it will live FOREVER. It cannot be destroyed. When a PID is merged into another or immediately "deleted" after it was created (and nobody else has touched it), it is ARCHIVED in the database, meaning that at any time it can be retrieved back into the FSFT if needed.
If a person record contains modifications by more than one person, the "delete record" function will never be available. But anyone can merge it with another person record (thus archiving one of the two PIDs that were merged).
also keep in mind many people perceive a duplication when there is more than one spouse of a person - and the screeen does show the person twice - once with each spouse - but each instance would have the same ID number - indicatin it really is not a record duplication.
That may have been rather confusing what AI jsut said - but . . .
For us to help you more precisley - can you share the PID of the person(s) so we can take a look.