Note: The pet in this story isn't always Mexican. The country of origin depends on the version and on the origin of the teller, but this is the most popular version in the US.
A couple were going to Mexico for a second honeymoon type trip, and obviously they didn't want to bring their ten-year-old son on this romantic getaway. But they promised to bring him something back so that he wouldn't be upset about getting left behind.
The last day of the trip, they shopped around for something to take him, but everything they thought he'd like was out of their price range. They got take-out food for lunch and took it to a park to eat it. As they ate, a cluster of stray dogs gathered around them, begging for scraps. One, a very small, scruffy, but cute dog, was particularly hungry, and followed them around as they began walking away. They decided it was adorable and their son would love it. So, they smuggled it over the border and took it home.
Their son did love it. After they got it home, they gave it a bath, got it some food, and played with it a bit. It seemed to be a quite satisfactory pet, though it wasn't as affectionate as dogs usually are.
For a week, things went on as normal. The dog was a bit standoffish, but it was cute and seemed happy in its new home. Then, one morning, the boy woke up to find it gnawing on his ear. He screamed and his mother came in and managed to get it off him. She realized that it looked sick and took it in to the veteranarian. He took one look at it, and asked her, "Exactly what kind of animal did you say this was?" She replied confidently, "It's a dog, some kind of rare Mexican breed." The vet shook his head. "Oh, no it's not. This is a sewer rat." The woman just about fainted. "A sewer rat?" she cried. "Yes," said the vet. "And that's the good news. The bad news is, it has rabies."