USB device with a serial number and 3 LEDs
You don't need an external EEPROM for this. Get a microcontroller with EEPROM instead, that's smaller and easier to use, because you don't need an SPI or I2C module anymore. Also, you can easily program the EEPROM from the computer.

So, basically, you want a PIC or other microcontroller with:


(E)U(S)ART connection
EEPROM
3 GPIO pins for the LEDs


Most microcontroller manufacturers have a part selection tool. For microchip, that's MAPS. That's how you can lookup which chip you need.

For these (very minimalistic) requirements, a very small chip enough. Have a look at the PIC12F1822: 8 pins, XLP if you need that, 256 bytes EEPROM, etc. This chip has just enough I/O pins for your three LEDs.

However, from your question it looks like you're a beginner in PIC, and then perhaps a somewhat bigger but easier to interface chip is a better start. Then have a look at the PIC16F88: it has everything you need and more I/O pins, for more LEDs, if needed. The fun of the 16F88 is that it's easier to program it. For the 12F1822, they put everything in a small box and you have to initialize everything (a lot!) correctly, which can be hard, especially for a beginner. The 16F88 is easier to start with.