What, if anything, does the phrase, "clear to maneuver" mean if given by an FAA airspace controller?
That isn't standard FAA phraseology; at least, it isn't anywhere in the ATC orders. The closest thing is in section 9-2-20 on evasive action maneuvers. I know nothing about them, but they seem to be a military training thing:
CLEARED TO CONDUCT EVASIVE ACTION MANEUVER FROM (fix) TO (fix),
(number of miles) EITHER SIDE OF CENTERLINE,
MAINTAIN (altitude) THROUGH (altitude),
COMPLETE MANEUVER AT (fix) AT (altitude)
But of course both pilots and controllers use non-standard phraseology all the time, especially for unusual situations. It's certainly possible that a controller somewhere has indeed said "cleared to maneuver as requested" or something similar. Although the 'correct' phraseology would be "approved as requested" (see section 2-1-18 of the orders) for a general request.
Another possibility is that it's military phraseology, not civilian. If you can link to a source or ATC recording then you might get a better answer.
It means, on the radar and from other information report available to the controller, it appears that no aircraft are in the way, and that the maneuver can be performed in time. No other aircraft appear to be coming in the area needed to perform the maneuver, in the time that the maneuver need to be performed. It also implies that the controller is aware and ready to respond in case of emergency.