I'm a sailor aboard the USS NIMITZ on the way home to San Diego.  We
are having a "Tiger Cruise," which is where sailors bring friends or
family members, called "Tigers," aboard to spend a few days
experiencing life on a naval vessel.  None of the 6,000 people aboard
this ship knows exactly how such cruises began, or why our Tigers are
called "Tigers."  What is the origin of this tradition and its name?

The only thing that I could find follows:

http://www.rso.navy.mil/chaps/tigercruise/tigercruise.htm

"The occasional embarkation of family members of Naval personnel is
intended to contribute to good morale and instill in servicemembers a
sense of pride in the Navy and their ships. It also enhances public
understanding of the Navy and increases community awareness."

"Operation Tiger Cruise is the unclassified code name for a very
special guest cruise program that includes two or more consecutive
days underway. The primary purpose of a Tiger Cruise is for
servicemembers to acquaint their family members with their ship and
their shipboard duties."

Additionally;

[PDF] CINCLANTFLTINST 5400.2M CH-3
File Format: PDF/Adobe Acrobat
Page 1. CINCLANTFLTINST 5400.2M CH-3 (N16) CINCLANTFLT INSTRUCTION
5400.2M
Subj: US ATLANTIC FLEET REGULATIONS l. Purpose. To promulgate ... 
https://www.nlmoc.navy.mil/cen/hurrex/ 2/CLF-5400pt2M-LANTFLT-Regs.pd

When I was in the Navy (1962-83), we called them "dependents cruises."
My guess is that the "tiger" in "Operation Tiger Cruise" was selected
arbitrarily in much the same way as certain other operation orders
(oporders) are named. I've emailed a couple old shipmates who might
know if there's a different answer and will post it here if I learn of
it.

nautico-ga, LCDR USN (Ret.)