Rotted bottom of wood siding

The 8" Home Depot siding will be too narrow. It's 8" nominal, the actual exposure width will end up under 7". To avoid the need to exactly match widths, consider replacing the boards full height, then you have no butt joints in the siding.

If you don't want to spend for the extra material, finding siding that exactly matches width may not be that easy. You could get someone to mill boards to a custom width. Since you boards have no special exposed profile, someone with a table saw could easily cut boards to match what you have. The raw material would just be nominal 1x12 and 1x10 lumber with one face resawn, which is a pretty common siding raw material.

This is a relatively easy DIY project, but it may take more than 3 hours. Be sure the siding nails you use are either hot dipped galvanized or stainless steel. Electro-plated galvanized nails will still rust. Siding nails have more of a head than finishing nails, so the boards are held tight and can't warp, but smaller than regular nails so the exposed head is not too objectionable.

If you want everything to match, replace everything. Otherwise, you will always be chasing trying to immediately match the existing as other pieces of siding degrade in the future.

Painting sounds like an option, but in this case, we're talking about painting old wood some of which is reaching the end of its lifecycle. That means in a few years trying to match weathered paint instead of weathered wood...so probably repainting again.

One alternative is to relocate existing weathered pieces from a less visible location to any prominent location and to install the new wood in the less visible location. It makes the job slightly larger, but still smaller than replacing everything, painting, or artificially aging the new material.

As a note, the concrete slabs/pavers adjacent to the siding is probably contributing to the speed at which the siding degrades because water dripping off the roof will have more 'splash energy' after hitting the hard surface.