What kinda of wood or other material should be used to replace the box end and soffit?
Looks like you've got 1x6 boards as your fascia which are probably pine, cedar or something similar depending on your region with plywood under the soffit. This is how they are traditionally done.
Nothing wrong with this but they do require maintenance. You'll pay a premium these days for solid wood which is why most builders and budget conscious people pick cement and man made products.
This honestly looks like it is decent shape still and is just due for some maintenance. But pictures can be deceiving. Poke it with a screwdriver. If it's solid then strip and repaint. It may take a little more labor if a board needs replacing but I personally favor repairing something that is fixable.
In Louisiana so I'm guessing pine? Top and middle boards are spongy at the corners.
Then I'd say pry those two boards out. Take them to your local lumber supplier and pickup some new stock that is the correct dimension. Some places carry pre-primed lumber to save some labor. Use the existing pieces as a template to cut your new stock to shape. Scrape or sand the fascia that is solid. Prime and paint it all. Don't forget to use some outdoor rated fasteners that won't rust.
Which species of wood to use is a whole topic in itself, but the more expensive options are naturally rot resistant like cedar and redwood. I think cypress? is native to your area so that may be what they stock locally.
With regular paint maintenance though most species will hold up well.
Fibre cement sheets would work and very cheap.
Treated timber Hx.y, I forget the minimum x.y you'll want for exterior cladding.
Would Home depot ProWood 3/4 in. x 4 ft. x 8 ft. Ground Contact Pressure Treated Pine Performance Rated Sheathing Plywood work?
Ground Contact Pressure Treated
Higher spec than you need but it'd be fine.
In my area azek PVC boards are commonly used. Not sure how well it would mean with the remaining trim though.
Home Improvement