To answer the statement that isn't a statement so much as a question in a mask in this context due to its first phrase being 'I think'...  **yes**, both of them can be rephrased into this form, **and this is more common and natural**. And more pleasant, for the blessed receiver of said cup of tea.

The difference between the 'for' and the 'to' here is (as far as I'm aware) the *verb*. Let's try replacing *bring* with *give*:

>Don't get up – I'll give a cup of tea to you.

>Don't get up – I'll give a cup of tea for you.

You should notice here that the second one is definitely *not* natural to say. This is simply because you *give* something *to* a person.

However, when you bring something, you *bring* it *for* them, because it is considered a *gift* in this context.

When you talk about *bringing a cup of tea*, the cup of tea is a gift and is thus brought *for them*.

...Final word: **using 'for' is certainly more natural** than using 'to'; though 'to' is correct I wouldn't recommend using it instead of 'for'. Then again, **I'd recommend you rephrase the sentence to: "I'll bring you a cup of tea", as you mentioned.**