Is there a rule on how many instruments a big band needs to have? Also, I was always wondering why do they call them Big Bands?

No rules but generally it was always 12. 
5 saxes, 3 trumpets, 1 trombone, piano, bass and drums.

The term "Big Band" was from the 1930's music era. As opposed to A trio etc.  Most bands also had at least a male or female vocalist.

In the late '30's and up to date, the bands grew to 4 trumpets, 4 trombones
because the arrangers wanted more orchestral depth and sounds.

There is no set rule but the standard "Big Band" has 4 trumpets, 5 saxes (2 altos, 2 tenors and a baritone) 4 trombones (3 tenor and 1 bass) 1 drummer, 1 guitar, 1 keyboard (usually piano) and a bass player.