Twelfth Night: Why does Olivia call Sir Toby “cousin”?

Solution 1:

Because the original sense of the word in English, per the OED, was:

A collateral relative more distant than a brother or sister; a kinsman or kinswoman, a relative; formerly very frequently applied to a nephew or niece.

Which also includes the following citation:

• 1599 Shaks. Much Ado I. ii. 2 ― How now brother, where is my cosen your son?

Solution 2:

A similar usage exists in modern-day slang (source):

cuz (n.) - cousin; friend; brotha; homey/homie

I have been called this by two of my uncles (thus adhering to the original sense), and it is used intermittently between family members. I am not sure how widespread the practice is outside of gang culture–which I'm not involved in–but I can attest to its usage in parts of the U.S. Midwest.