What is a good project for amatuer PCB designers?
TL; DR - PRACTICE and get your design reviewed by someone who is in the industry for long. That's how you will improve. 

I'll share my story. I'm still amateur at PCB design; although have fared well in past year.

I started in 2009 when i was in my 9th grade. I desperately wanted a MP3 player. My father didn't want to throw money on expensive iPods(ipods were the new cool at that time). So I took it to myself; built one myself. Of course with help. I didn't have the technical know-how of any design. I read books like crazy. Couldn't comprehend a word. 

In this process; because i was short on money; I decided to fabricate a PCB myself. I discovered options like toner transfer that is possible at home. Eventually, now, after 7 years i can manufacture good pcbs for initial prototyping. 

I got the PIC18F series programmed by someone who was 500 miles away. I was 15 at that time. 

Completed this project when I completed my 12th grade. About 3 years. 

TECHNICAL COMPETENCE = ZERO. 

But, it worked. 
I started my Bachelors in Electronics and made amendments to the existing design. PCB Designs got better and progressed as time passed.  In  the final year of my bachelors, I am pushing good designs which find their place in industrial components. From single layer to six layer boards with RF Components.  I've picked up skills from forums, tutorials, electronics.stackexchange. Olin Sir had amazing answers. But one thing remained common, PRACTICE and learning from mistakes. There's no shortcut.