What timber sizes do you recommend for doing a light structured lean-to roof over a deck?
It is hard to gauge the size of the timber that is needed without knowing the full extent of your idea, but NZ Wood has a general guideline for building a light structured lean-to roof (pergola)
First we would suggest that you check with your local council to see if you need building consent (especially if you are thinking of attaching it to your house). They are able to give you a quick rundown on the building code and if you need to apply for building consent.
To give you an idea of what timber and materials are recommended for use, view the how-to-build-guide for building a free-standing pergola which gives you an idea of what timber sizings, treatments and materials are recommended for use.