Use sp_rename with type:

EXEC dbo.sp_rename 
   'dbo.FK_FRP_File_DataMatrix',
   'FK_FRPFileID_FRPDataMatrixImportETL',
   'OBJECT';

Demo:

CREATE TABLE tab1 (ID INT PRIMARY KEY,
                    col1 INT NOT NULL)
CREATE TABLE tab2 (ID INT PRIMARY KEY)

ALTER TABLE tab1
ADD CONSTRAINT FK_FRP_File_DataMatrix FOREIGN KEY (col1)
REFERENCES tab2(ID);

 EXEC dbo.sp_rename 
   'dbo.FK_FRP_File_DataMatrix',
   'FK_FRPFileID_FRPDataMatrixImportETL',
   'OBJECT';

SELECT CONSTRAINT_NAME
FROM INFORMATION_SCHEMA.REFERENTIAL_CONSTRAINTS;

SqlFiddleDemo

Thanks but I get this error "Either the parameter @objname is ambiguous or the claimed @objtype (OBJECT) is wrong.". I believe the issue is some how when I created the foreign key the Schema name 'dbo.' is actually part of the name of the object. So when I exec the sp_rename it looks in Schema dbo for FK_FRP_File_DataMatrix which doesn't exist. The name really is dbo.FK_FRP_File_DataMatrix in the objects table. If I manually remove the dbo. from the name under Keys then and rename it. Need to fix this on my production database.

Drop the constraint on the old table ( Table1_Old ):

ALTER TABLE dbo.Table1_Old DROP CONSTRAINT FK_Table1_Table2
GO

Add the constraint with the new name on the old table:

ALTER TABLE dbo.Table1_Old  WITH CHECK 
ADD  CONSTRAINT FK_Table1_Table2_Old FOREIGN KEY(FileId)
REFERENCES dbo.Table2 (FileId)
GO

Now you are free to add the constraint on the new table ( Table1 ) with the old name of FK_Table1_Table2 .

I ran into this error today. To add to the currently upvoted answer, you need to include brackets around each piece of the tablename like so:

sp_rename '[dbo].[table_column]', 'newvalue'