I have a client moving to O365 just now, and I had considered building it as a hyrbid on-prem and cloud system, figuring there might be some benefit. I can't find much online though, other than from the point of view of already having an O/P install you want to shift to the cloud.
Two things I was hoping might be possible with hyrbid were... (and forgive my naivety if these seem stupid, I'm just spitballing, pretty sure they're not doable as MS are far too clever to let someone have a get-out of the tri-lock they put on the engine room with 365)
Admin - if you have hybrid, do you get any more admin access to things like timer jobs, ULS etc? I'm guessing not, but this is my biggest peeve with 365 as it makes things like crawls and content source publishing quite ambiguous, especially if you're on a tight deadline and have to tell the client 'er, it'll be however long MS decide it'll be'
Extranet - the client wants some external users (around 50) to have access to a document center used to store tech manuals, procedures, signoff sheets etc. They would probably be fine with using anonymous sharing but if the external users had full access they could do so much more with it - client probably wants to avoid paying for extra licenses tho, even Kiosk ones. If you had an O/P server as part of a hybrid, could you use Sharepoint Foundation (2013, obviously) to host an extranet specifically, meaning there'd be no license costs? We have another client that does something similar with Documentum and SP2010 Foundy, just not sure if it'd work with hybrid as the site would still count as being in 365, I'm guessing