Final answer:
The organelle that enabled early eukaryotes to synthesize food and produce oxygen is the plastids, specifically the chloroplasts which perform photosynthesis.
Step-by-step explanation:
The organelle that may have allowed early eukaryotes to make food and produce oxygen is the plastids. Specifically, the chloroplasts within the plastids are responsible for photosynthesis, the process by which plants and plant-like cells convert light energy into chemical energy, thus producing food in the form of glucose and releasing oxygen as a byproduct.
According to the Endosymbiotic Theory, proposed by Lynn Margulis, chloroplasts originated from endosymbiotic cyanobacteria that were engulfed by primitive eukaryotic cells. Over time, these bacteria developed a symbiotic relationship with their host cell, evolving into the organelles we now know as plastids, including chloroplasts. This is evidenced by the fact that chloroplasts have their own DNA, similar to cyanobacteria, and have a double membrane, a trait of gram-negative bacteria.