Translation:He said that he will give me a toothbrush.
Don't worry, I added it as an accepted answer the moment you mentioned it.
I see this the same way as "I didn't know you were a doctor" versus "I didn't know you're a doctor". That is, the first sounds a bit more natural, but the second isn't wrong if you really want to avoid any confusion. That is, if you don't want to suggest that s/he is no longer a doctor.
So in this case, "He said that he would give me a toothbrush". Could mean that he gives you the toothbrush either before or after the moment of uttering the sentence. "He said that he'll give me a toothbrush", only means that he gives you the toothbrush after the moment of saying the sentence.