Try this
CREATE PROCEDURE usp_TestLike @VarName varchar(100)
AS
SET NOCOUNT ON
SET @VarName = ISNULL(@VarName,'') + '%'
SELECT *
FROM sysobjects
WHERE [name] LIKE @VarName
GO
This will work with the variable that start with this name if you want to look for variable in any portion then change the SET to
SET @VarName = '%' + ISNULL(@VarName,'') + '%'