How to fix window that won't stay open/what is this part called?
This is the modern equivalent of a sash weight.

Yours is either a (window) spring balance or a spiral balance.

But you seem familiar with the terminology, so the real question is: what's that little plastic part called? 

It's either a Nylon Sash Cam or a Shoe Set I think.

It might just be easier (and better) to purchase a whole Window Channel Balance Repair Kit and replace all the plastic parts. They're like $5. Then again, an entire balance costs like $10.