What does Leviathan mean?
Hebrew livyathan, a great sea-serpent/monster, generally glossed 'twisted, coiled' and derived (per Gesenius) from the root l-w-h 'to twine, wind' with a suffix, i.e. 'wreathed one'. Flag: the derivation is disputed (some hold it a loanword), and in use it is the name of a biblical serpent-monster; the 'coiled serpent' sense is the established reading.
What does Shai mean?
From Hebrew shai 'gift'. The 'gift' meaning is widely cited, though the origin is disputed: some sources derive the name instead as a diminutive of Yeshayahu (Isaiah), 'God is salvation'.