Just to let you know, "Let him have it." is also an idiom, meaning to get angry, etc. at a person. EX; "I told him to stop bothering me, or I was gonna let him have it." This usage may be becoming slightly antiquated, but it's still in use.

It can mean different things depending on the context. Please give context in a story.

He asked you for something and you agreed to give it to him. The thing in question could be anything. 'He asked if he could use the computer at 5pm.' 'I let him have it.' 'She wanted the time off next Wednesday to see her son.' 'I let her have it.' Another, more formal, way to say this would be 'I agreed to /accepted his/her request.'