means To walk or move unsteadily, as if about to fall — or, by extension, to space things out so they don't all land at once (staggered start times, staggered shifts).

from From Old Norse 'stakra,' meaning to push or stagger, a frequentative of 'staka' (to push, totter). It came into Middle English as 'stakeren,' and the spelling drifted to 'stagger' over time. The two-thing sense — spacing events apart — is a much later figurative leap: things arranged so they lurch into view one after another rather than all at once.