From Middle English bicasten, equivalent to be- + cast. Sense of plan, plot due to conflation with costen (“to contrive, try”). More at costen.
becast (third-person singular simple present becasts, present participle becasting, simple past and past participle becast)
- (transitive, obsolete) To cover or surround by casting (something) about.
- (intransitive, obsolete) To cast about; plan; plot; contrive.