For someone with sensitive skin, I would recommend the following materials for a green throw blanket:
Bamboo
Bamboo fabric is soft, breathable, moisture-wicking and hypoallergenic. It’s a great choice for sensitive skin.
Microfiber
Microfiber is very soft, quick-drying and less likely to harbor dust/allergens than natural fabrics. Good option.
Alpaca
Made from alpaca wool, it’s extremely soft and breathable. Many find alpaca warmer than they expect.
Cashmere
Incredibly soft and lightweight, cashmere regulates temperature well. However, it’s very expensive.
Tencel
A sustainably produced fabric from trees, tencel is smooth, breathable and less irritating than other fibers.
Linen
Made from flax, linen is breathable, moisture-wicking and milder than cotton. Great for warmer weather.
I would avoid wool if the person has known wool sensitivities. Cotton can work if a loose weave like flannel, but rough terry cloth should be avoided. Look for dyes/finishes labeled “sensitive skin safe” too. Bamboo, microfiber or tencel would give the softest feel with low irritation risk.