My poor dog Tray

In old literature, a common name for a dog is Tray. What’s the origin of this?

Brewer, in his Dictionary of Phrase and Fable says that it either comes from trag, a word meaning a runner, or from the Spanish traér, “to fetch.”

He did not specify a language for trag and I did not find it in the OED (under trag or under tray).