1
general::
Attend to local or family matters, as in Ask Dad for permission; he's minding the store while Mom's away. This expression transfers looking after an actual business to more general activities. [Colloquial; second half of 1900s]
American Heritage Idioms
2
general::
Phrase(s): mind the store [and] watch the store
Fig. to take care of local matters. • Please stay here in the office and mind the store while I go to the conference. • I had to stay home and watch the store when Ann went to Boston.
McGrawhill's American Idioms And Phrasal Verbs