Lab vs personal brand
Unless you are 100% that you will run this lab until you retire (and even then, plans can change), I do think it makes sense to keep both websites and decouple them a bit. You and the lab are two different entities, possibly with different relevant recent achievements to publicize, and in any case, presumably you’ll want to grant employees/students of your lab access to modify the lab website, but probably not to modify your personal website!

For your personal website, I’d include:


material for courses taught
personal research interests
your CV
link to your lab page
etc.


For the lab website, I’d include:


lab publications
lab projects
recruiting info
list of personnel and links to their personal pages (including yours, of course)
news/media coverage/events
etc.


As for pros &amp; cons, I’d say the pros of distinguishing you vs the lab are:


enables personnel to feel more “ownership” in the lab, thinking of themselves as part of an organization rather than just as your employee
I actually think a lab named for what it does (e.g. “Climate Dynamics Lab”) is not “generic” but actually more transparent and meaningful for communicating what your lab group studies (at least until you become so famous that your last name becomes synonymous with your subfield)


Cons:


you’ll have to maintain two slightly different websites, so synchronization becomes a pain