The British arrival into the Pacific Northwest in 1792 was a very big deal, for many, many reasons. Chief amongst them? The names of many of the places that we take for granted today had their origins with the British Captain who led his ship into the region.
George Vancouver was the captain of the HMS Discovery, a 10-gun survey ship, who was tasked with wresting control of the Pacific Northwest from the Spanish, either by diplomacy or by force. The Discovery used this opportunity to explore the Pacific Northwest, in hopes of better establishing their claims to the region.
Vancouver used this as his impetus to start claiming and naming everything in sight, with hopes of insinuating the British and British influences into the region. The Puget Sound was so named due to the existence of 3rd Lieutenant Peter Puget, who served beneath Captain Vancouver. Puget explored the waters of the area in longboats from the Discovery, while Captain Vancouver busied himself with other duties.
The name "Puget Sound" was initially meant only to refer to the region of water south of Tacoma, with the waters north of present-day Tacoma to be called Admiralty Inlet. The latter name fell out of disuse, and Puget Sound soon represented all of the waters south the the Port Townsend and Fort Casey.