For a construction site, you need a trash bin that's tough, simple, and gets the job done without fuss. I'd suggest steering clear of anything designed for home or light office use. You want something built for heavy-duty, industrial environments.
My top recommendation would be a basic, open-top steel dumpster or roll-off container. These are the ultimate in no-frills functionality. They're incredibly durable, can handle heavy debris, and are easy for crews to toss materials into. For smaller sites or more mobile needs, a heavy-duty polyethylene or galvanized steel mobile trash cart is a great choice. Look for one with large, sturdy wheels to move over rough terrain. Avoid bins with complicated lids or foot pedals—these often break on site. The key is simplicity, capacity, and durability. A plain, robust bin will withstand the elements and job site abuse far better than a "feature-rich" model.