SQL Server 2005 onward, we can create the roles in database. You can follow below simple script to grant the EXECUTE permission to user.

-- creating the database role 
CREATE ROLE Database_Executor
-- granting the execute permission to database role
GRANT EXECUTE TO Database_Executor

-- Here I am granting the SQLDBPool login execute permission
USE [DBName]
GO
CREATE USER [SQLDBPool] FOR LOGIN [sqldbpool]
GO
USE [DBName]
GO
EXEC sp_addrolemember N'Database_Executor', N'sqldbpool'
GO