Are cotton bags the most eco-friendly?

No, that's a myth:

• LCA analysis: Life-cycle studies (e.g., from the Danish Ministry of Environment) show that cotton, especially organic, has a high environmental footprint—requiring 1600–10,000 uses to offset a single plastic bag.
• Why: Intensive cultivation with high water, energy and fertilizer use.

Synthetics (e.g., polyester) are often more eco-friendly with far fewer uses.