I like DC's suggestion of "Rex." Not only does it sound like "Lex," it means "king" in Latin.
My ideas:
1) Cerberus
2) Depending on when he named the dog vs. when he learned about a certain green rock, he might name the dog "Krypto," short for "Kryptonite." (For those who haven't read the comics, Superman had at one point had a dog named "Krypto," so there is an irony here of which Tempus would be proud.) Also, the prefix crypto means "hidden" or "secret."
Joy,
Lynn
