The earliest cell is thought to have emerged at least 3.8 billion years ago at a time when the environment was anaerobic in nature (1-2). It is suggested that simple organic molecules formed and gave rise to a self-replicating RNA that found itself within a phospholipid membrane (2).
The evolution of metabolism began with glycolysis (2), the sequence of reactions would produce life's universal energy source: ATP (1-3). Photosynthesis and oxidative phosphorylation would come later (2).
While cells may not need certain parts to remain living, all still depend on glycolysis to generate energy (2). The pathway is contained within the cytoplasmic matrix (1-2). Thus, I propose the cytoplasmic matrix is the most important part of the cell.
References
1. Dennison KJ, Topping J, Caret RL. General, Organic, and Biochemistry. New York: McGraw-Hill, 2007.
2. Cooper, G.M. "The Origin and Evolution of Cells." The Cell: A Molecular Approach. Available at: http://www.ncbi.nlm.nih.gov/books/bv.fcgi?rid=cooper.section.90.
3. Gropper SS, Smith JL, Groff JL. Advanced Nutrition and Human Metabolism, 5th ed. Belmont, CA: Thomson Wadsworth, 2009.