Name for wooden board that supports fascia, drip edge, soffit, and gutter?
What you see exposed in that picture is the fascia board. There is usually a fascia cover board on most homes as well. However, your fascia is covered with siding. I would recommend determining what is actually causing the rot and repairing that first. I see a huge issue in your photos. Your shingle overhang is way too short in places. It appears that some places have no overhang at all. Your shingles need to hang at least 1/2"-3/4" past the drip edge. When shingles are not hanging past the drip edge, the water will simply run over and under the drip edge and into your fascia material below.