Tabela MySQL Inner Join com base no valor da coluna

Para ter as estatísticas de todas as tabelas, você pode usar um UNION, com 2 ou mais selects, um para cada tabela:

( SELECT s.*
       , table1.title AS name      --or whatever field you want to show
  FROM stats s
    JOIN $tableName1 table1
      ON s.id = table1.id
  WHERE tableName = '$tableName1'
)
UNION ALL
( SELECT s.*
       , table2.name AS name      --or whatever field you want to show
  FROM stats s
    JOIN $tableName2 table2
      ON s.id = table2.id
  WHERE tableName = '$tableName2'
)
UNION ALL
( SELECT s.*
       , table3.lastname AS name      --or whatever field you want to show
  FROM stats s
    JOIN $tableName3 table3
      ON s.id = table3.id
  WHERE tableName = '$tableName3'
)
;

Usando a ideia de Winfred com LEFT JOIN s. Produz resultados diferentes, e. cada campo das outras tabelas é gerado em sua própria coluna (e ocorrem muitos NULLs).

SELECT s.*
     , table1.title      --or whatever fields you want to show
     , table2.name
     , table3.lastname   --etc
FROM stats s
  LEFT JOIN $tableName1 table1
    ON s.id = table1.id
      AND s.tableName = '$tableName1'
  LEFT JOIN $tableName2 table2
    ON s.id = table2.id
      AND s.tableName = '$tableName2'
  LEFT JOIN $tableName3 table3
    ON s.id = table3.id
      AND s.tableName = '$tableName3'
--this is to ensure that omited tables statistics don't appear
WHERE s.tablename IN
   ( '$tableName1'
   , '$tableName2'
   , '$tableName3'
   )
;