Travel
Lonely Planet Solomon Islands by Mark Honan and David Harcombe — the standard travel guide, though editions go out of date quickly given how much changes on the ground. Still useful for context and island geography.
History and WWII
Guadalcanal: The Definitive Account of the Landmark Battle by Richard B. Frank (ISBN: 0140165614) — the best single-volume account of the 1942-43 Guadalcanal campaign, covering the naval battles of Iron Bottom Sound, the land fighting, and the strategic stakes. Essential reading before diving the wrecks.
South Pacific Destroyer: The Battle for the Solomons from Savo Island to Vella Gulf by Russell Sydnor Crenshaw (ISBN: 155750136X) — a naval officer's account of the destroyer battles in the slot, readable and specific.
Culture
Living Tradition: A Changing Life in Solomon Islands by Michael Kwa'iola and Ben Burt (ISBN: 0824819608) — a close portrait of life in Malaita, combining personal memoir with ethnographic observation. One of the better books for understanding how kastom and modernity coexist.