The historic Gump Building was designed by Hart Wood. Hart Wood designed a lot of other buildings worth staring at from both afar and up close. Alexander & Baldwin (in collaboration with Dickey) and Spalding House (although several other architects have worked on it over the years) are just two examples. The building was constructed for Gump's, a store founded in San Francisco in 1861 by Solomon and Gustav Gump, resellers of art collected on their world travels. There were also Gump's stores in Los Angeles and Carmel. The building opened in 1929, making it the third oldest building currently in Waikiki (after the Moana Hotel (1901) and Royal Hawaiian (1927); as far as I know). Here is a description of the building from the 1969 Historic Buildings Task Force taken from "Old Honolulu: A guide to Oahu's Historic Buildings":
"Combining Oriental motifs with Hawaiian, the architect used imperial blue roof tiles, exterior woodwork of stained Burmese teak, and verde antique copper gutters. Within, Oriental rooms were appointed with hand-crafted Japanese tatami mats and shoji and Chinese hardware. Display walls were covered in carpet; concrete floors were acid-stained in mellow earth tones and greens for an aged effect. The ceiling of the Spanish Room was gray weathered wood; other ceilings were left with their rough form marks and stained."
Looking at it today, it sounds as though the exterior remains very much the same. Once inside, though, it is hard to tell you're in a historic building at all. The Asian accents have been replaced by a modern showroom with glossy surfaces and dropped ceilings (at least on the first floor; the second floor ceiling is open with exposed beams).
For a while, possibly in the 1970s or 80s, McDonald's set up shop here. Now the home of Louis Vuitton, the building's use is closer to that originally intended.
Louis Vuitton, like Gump's, has a rich and storied history. The company was founded in Paris by its namesake in 1854, just seven years before Gump's. LV was originally known for its trunks with innovative stackable, lightweight, and airtight flat tops and bottoms. You can see some of their early trunk designs on their website.
More info: https://en.wikipedia.org/wiki/Gump%27s