to allot (a task, place, accommodation, etc) to somebody
to appoint (somebody) to a particular task, post, duty, etc
to transfer (property) to another person, esp in trust or for the benefit of creditors
to specify or fix authoritatively (a date for a future event, etc)
to ascribe (something) as a motive or reason
assignable adj
assigner noun
assignor noun
[Middle English assignen via Old French from Latin assignare, from as- + signare to mark, from signum mark, sign1]