What about having an earlier Gold Rush? In OTL Gold from the Gold Ridge mine only began to be exploited in the late 1930s. Seeing as this was posted in the Post-1900 forum, perhaps the same Gold is discovered in 1900 resulting in a large influx of miners, prospectors, and various hangers on. This probably isn't enough to base an economy on, but it could be the prompt for the beginnings of some infrastructure. Say that the British then proceed to build a naval base in the region to further hem in the Germans in Papua New Guinea.
Though most of the prospectors end up leaving Guadalcanal, enough stay, intermarry, and invest to create the beginnings of a local economy and insures that the city founded to house all the prospectors survives. As the South Sea Islanders are deported following the Pacific Island Labourers act of 1901, perhaps a number make their way to Guadalcanal and settle there. Primary industries are of course whatever mining continues, fishing, as well as providing supplies to the local RN base. Tropical wood which is the mainstay of their exports in OTL is another good avenue for early expansion. Perhaps, rather than a racist hierarchy, a more inclusive melting pot emerges among the Non-RN inhabitants of the islands and the beginnings of a local culture start to take shape. An enterprising missionary essentially creates a language that's close enough to all the local dialects to be mutually intelligible and translates the Bible into it as well as other media aiding in cultural cohesion.
Perhaps during the 1930s and 40s the Solomon Islands becomes a haven for Jewish refugees fleeing Europe? The majority settle on Guadalcanal and then come to play a key role in the war effort as the island is reinforced to keep it out of Japanese hands and then besieged during a long and bloody Solomon Islands Campaign. While Allied and Japanese troops fight over the northern islands in the peninsula, Guadalcanal becomes the Pacific version of Malta, complete with a medal being given to all of its inhabitants. As a key logistical point, the local economy also begins to grow by leaps and bounds, perhaps with some small scale manufacturing taking place. Wartime experiences further bind together the peoples of the Solomon Islands and even manage to convince a number of the Jewish refugees to stick around. One refugee in particular blends the island's tropical lumber industry with modernist design and excellent marketing, and ends up founding a furniture empire that rivals IKEA. Wartime experiences of resistance and solidarity help serve as the basis for an emerging national identity that pushes for independence after the war and eagerly welcomes refugees.
Following the war, one of the presidents of the US, when recounting his service in the Solomons publicly acknowledges his rescue by some islanders from certain death at the hands of the Japanese. This along with what
@Insider mentioned could lead to a tourism boom. Furthermore different cultural developments in TTL avert the inter-ethnic violence that occurred in OTL towards the end of the 20th century.