public void SetName( 
   string attachmentName
)

public async Task UpdatingAttachments()
{
  await ArcGIS.Desktop.Framework.Threading.Tasks.QueuedTask.Run(() =>
  {
    using (Geodatabase geodatabase = new Geodatabase(new DatabaseConnectionFile(new Uri("path\\to\\sde\\file\\sdefile.sde"))))
    using (FeatureClass landUseCaseFeatureClass = geodatabase.OpenDataset<FeatureClass>("LocalGovernment.GDB.LandUseCase"))
    {
      QueryFilter filter = new QueryFilter { WhereClause = "CASETYPE = 'Rezoning'" };

      using (RowCursor landUseCursor = landUseCaseFeatureClass.Search(filter, false))
      {
        while (landUseCursor.MoveNext())
        {
          using (Feature rezoningUseCase = (Feature)landUseCursor.Current)
          {
            IReadOnlyList<Attachment> rezoningAttachments = rezoningUseCase.GetAttachments();
            IEnumerable<Attachment> filteredAttachments = rezoningAttachments.Where(attachment => !attachment.GetName().Contains("rezoning"));

            foreach (Attachment attachmentToUpdate in filteredAttachments)
            {
              attachmentToUpdate.SetName(attachmentToUpdate.GetName().Replace(".pdf", "Rezoning.pdf"));
              rezoningUseCase.UpdateAttachment(attachmentToUpdate);
            }
          }
        }
      }
    }
  });
}