Definition
To think carefully and slowly about something before making a decision. Often used with "over."
Example
She mulled over the job offer for several days.
She thought carefully about the job offer for several days.
Etymology
From Middle English *mullen*, meaning to grind or crush into powder — the idea being that you "grind" a thought over and over until you reach a conclusion. The same root gives us the spiced-wine sense: *mulled wine* is wine that has been heated and steeped (thoroughly worked over).
Cultural note
*Mull over* is the standard everyday phrase; dropping "over" sounds slightly more formal or literary ("I need to mull this"). British and American English both use it equally. Unrelated to the Scottish word *mull* (a promontory of land, as in the Isle of Mull).