A sensor to recognize my pet near me
Off topic for Arduino, but you could put a collar with an Apple iBeacon on your pet and implement an iOS app that would wake up and display a local notification (optionally including a sound) when your pet comes near.

You could probably do something similar using BLE, but Apple's iBeacon has the concept of geofences based on proximity to one of their iBeacons, including distance ranges from "immediate" (about a meter) to far (30+ meters, if memory serves). That's actually somewhat tricky to do using vanilla BLE.

I think in order for this to be practical you will need to put some sort of transmitter on your pet. The RFID chips that vets put into pets require that the sensor be within a few mm of the chip, so that probably wouldn't work. (From what I've seen you have to wave a wand directly over the chip on the pet's skin in order for it to be detected.) There are other RFID technologies that should work. I found this link from a Google search, for example: 

https://skyrfid.com/RFID_Range.php