Kind of obvious question (unseasoned gardener, here), but wouldn't it be an idea to get some shade on that part of your house? A good deciduous tree might make a great difference in your options for that bed, not to mention your air bills.

The problem you are going to run into is that 2.5' is simply not deep enough, unless you are willing to attach a couple of trellises to the wall, and even then....Is there an eave/overhang over this area that keeps rain from hitting the ground? If so I don't think there's much you could do.

Roses might do well in a spot like that. They like lots of sun and if you planted climbers you could keep the bed narrow, say 5'.

No matter what sun loving things you plant there you're going to need to do something about the dirt in preparation for planting.

One of these vines on a trellis would tolerate your sunny location.
If you decide to use a vine, I would locate the trellis as far from the house as possible, to clear the overhang.
A lattice trellis can be constructed from 1"x2"x6' lumber and mounted on fence posts, either wood or steel. Treat the wood with a preservative or 2-3 coats of marine varnish (or polyurethane) before attaching the vine. I would install a drip irrigation or soaker hose to provide water, until any plant you install, become established.

Confederate Jasmine (Trachelospermum jasminoides 'Madison') - Very fragrant!

Mandevilla (available in several colors)

Rb

Some good ideas, and I just wanted to underline trees or something like those blanketing vines? The best heat control stops the sun rays before they actually hit the sides of the house. Espaliered pyracantha is another one that, once established, would handle the heat and drought, provide a modest flower show in the spring, green around the year, and bright berries in autumn and winter. It's thorny, though, and would need to be kept tidy. Our next-door-neighbor in California, though, had a gorgeous one on the wrong (away from us) side of his garage.