This weekend's German grand prix is actually the team's and Nico Rosberg's home race, but Briton Brawn thinks countryman Hamilton "deserves" his breakthrough.
Rosberg won Mercedes' first race of 2013 at Monaco, and he followed it up with the win at Silverstone last Sunday -- but only after local hero Hamilton suffered an exploded Pirelli.
"He (Hamilton) deserves a win," said Brawn.
Hamilton, 28, has had a solid first half-season with Mercedes after switching from McLaren, but he has often struggled with the handling of his all-new car.
It means Rosberg has generally had the upper hand, creating the impression Hamilton - who is reported to have broken up with his long-term girlfriend Nicole - is now desperate to win.
"It (winning) will give him a lot of motivation," agreed Brawn, "not that he needs it. But I think it will give him a boost to know he had the quickest car out there."
Brawn said Hamilton was "stronger" overall than Rosberg at Silverstone, "and I think Nico agrees".
Rosberg, however, is enjoying his position for now.
"Slowly but surely we are gaining the status as favourites," he is quoted by Spain's EFE news agency, "and it's fantastic."
The German also played down the effect Pirelli's tyre shakeup will have this weekend.
"No matter what tyres we use," said Rosberg, "we have a fast car and I think we can stay where we are."