had better (modals)
 

I often use “had better” to give strong advice, but my American friend told me that I was not so polite when I said, “You had better come over to my house for dinner.” Can you help me to use modals politely?
Thank you! :)

Re: had better (modals)
 

You were right when you said that had better is strong advice. The problem here is that in this situation "strong advice" is not appropriate - you are almost ordering your friend to come to dinner. had better is very close in force to should - it means the speaker is making a recommendation based on his/her own opinion, and will disapprove if the listener decides not to do it. For instance :
A : I'm getting really bad pains in my chest.
B : Oh no - that could be serious - you'd better go straight to hospital.
A: No, I hate doctors. I'll wait and see if they go away.
B : That's stupid!
In this case you have no right to "order" your friend to come to dinner, you need to invite him :
Would you like to come to dinner?
Do you want to come to dinner?
which allows him a free choice depending on what he personally wants to do. At most you could phrase it as a suggestion - "weak advice" :
Why don't you come to dinner?
How/What about coming to dinner?
Let's have dinner together.
Here the implication is that you think it's a good idea, but as it's only "weak" advice you won't be offended if he says no.

Re: had better (modals)
 

It's probably in the delivery, but I can see saying that to a friend. It would sound like an order but in a joking manner.

A: You'd better come over for dinner this Friday. If you blow me off again, I'm gonna get you.

They would need to be a close friend.

Other possibility is you've been giving help or advice on something possibly over the phone or email, but it's getting too complicated to explain over a distance.

A: You'd better come over for dinner. We can get everything done pretty quickly that way.

If you're just asking your friend or acquaintance over to dinner, I agree with Susan53's advice.

- It'd be great if you could come over for dinner. How does Friday sound?

- I'd love to have you over for dinner. When are you free?

For strong requests try adding "and I won't take no for an answer."

- I'd love for you to come over for dinner, and I won't take no for an answer.

Re: had better (modals)
 

Had better is often used as a warning or reprimand. To invite someone, you would use:

Why don't you come over for coffee later on?