Primary flight controls will be combination hydraulic/electric. Engines are rumoured to be sans hydraulic pumps. Not sure about flaps primary drive, but read somewhere that there is an integrated electric motor that varies the trailing edge shape of the wing for cruise drag reduction. Try a search on Parker Hannifin (sp?), Hamilton Sundstrand and Moog for more on the 787 hydraulics.
Electric braking sounds very interesting. Some sort of motor driven screw actuator would be my guess since a pure magnetic caliper would ,while quick, be heavy given the forces required. Goodrich has the contract for this item.