We’re are the results of mutations

My boy tried to make friends with a rat that was under my car, they were about 3 inches nose to nose. My boy looked me then at the rat again like , "can I play with him", then the rat got bored and started to slowly amble away under my car, my boy gave up then. lol

She learnt it from the cat, clearly!

It was what they were originally used for. "the Chihuahua is actually a fearless hunter of smaller prey, and in their native country of Mexico, they are prized as ratters, for hunting and killing vermin including squirrels and other small prey, especially rats and other rodents! Even within the domestic environment, the Chihuahua retains a strong hunter-killer instinct, and may display a significant amount of interest in domestic vermin!"
Most of the smaller agile dogs were bred for this purpose, much better at it than cats, and even the larger terriers are very keen on it. Beat cats hands down.

I have many different thoughts....

Dogs are very needy. It might be that the dog wanted to show it could also be a valuable hunter as a way of upping status in the pack in the eyes of hte Alpha (you).

It may also be tht the cat killed a mouse and left it, and the dog later found it, and brought to you to please you as he had seen the cat do many times.

Dogs have been bred to utilize natural huntign instincts to serve men. It did ge t6o the point where soem dogs will instinctualy brign prey to the masters.

Some breeds of dogs (such as terriors) were specifically bred specifically to hunt and kill rats and other rodents.

Maybe Lucy was taking credit for Hella's prey?

My little mix Pointed...yes, full hunting-dog point, the kitchen cabinet, making me aware that a mouse was attempting to move in. Would not stop despite me holding the door so he could go outside, his favorite thing. Totally focused!

I have Rat Terriers and Toy Rat Terriers, (The toys are the size of Chihuahuas, lol). A lot of small dogs were bred and used for hunting rats and mice and since they hunt to scale it works out great, makes perfect sense why your little dog is going for mice. Embrace it, you don't want the things in your house. I found by experience that my little dogs are far better ratters than any cat. Plus, they shake them and kill them instantly, where a cat will torture and play with them for hours, it's much better. And hunting stimulates them and makes them happy.

Probably jealous of the cat bringing home food so she thought I can do that too, I'm serious.

Simple prey drive