The name 'Bridgestone' is a literal English translation of the founder's name, Shojiro Ishibashi. Ishi= stone, and bashi=bridge which I thought was rather clever and possibly hinted at early ambitions in the export market. I also find it a bit sad that on the Bridgestone corporate web site history (see this link), there is no mention at all about their motorcycles, but rather just their tires. They do make nice tires, but they also made some really nice motorcycles, of which one would think they would still be quite proud.
Bridgestone had been in the rubber and tire business since 1935 under the Asahi Corporation name and was quite successful throughout Asia. Immediately following WWII, their overseas business was gone and their offices in Tokyo bombed out, but they did still have intact tire manufacturing capacity which allowed them to resume production immediately after the ending of hostilities. Recently, Bridgestone bought Firestone making them the world's largest tire manufacture so clearly they have done well. Bridgestone introduced a line of bicycles in 1946 (which I think they may still make) and then engine powered bicycles in 1949 using a clip-on engine sourced from Fuji Seimintsu Joygo (which is now a part of the Nissan group of companies). Actual motorcycles were introduced in 1958 beginning with the 'Champion' which was a 50cc 2 stroke with pressed steel frame and 3 speed gearbox, and a fully shrouded engine that was fan cooled.
The USA distributor was Rockford Motors in Illinois, initially via catalogue sales through Aldens of Chicago. The 'Champion' line was sold as the 'Super 7' and as well a step through was offered called the 'BS-50 Homer'.
The late 1950's and early 1960's saw many Japanese motorcycle manufacturing companies fail and Bridgestone was able to take advantage of the bankruptcy of Lilac and Tohatsu. The infusion of skilled engineers and technology from both failed companies resulted in a new Bridgestone product line with a number of innovative features (chrome plated cylinder bores and rotary valves for example) in 1964 and this product line largely continued unchanged till the end of production.
Two popular models sold exported globally were the 350 GTR which was a 37hp, aircooled 345cc 2 stroke twin, and the DT175 20 hp low pipe, and the 175 Hurricane scrambler version with high pipes and the same engine specifications. 100cc, 90 cc, 60cc and 50cc models were also sold.
While successful, Bridgestone motorcycles were expensive relative to their competition, and it appears that the tire sales side of the business was more lucrative than the motorcycle side, so a decision was made to end motorcycle production in 1970 or 1971. Manufacturing rights for the 60cc and 100cc mini-bikes were sold to a company in Taiwan and these continued to be imported into the USA as Rockford 'Chibi, 'Taka' and 'Tora' till 1975.
There are many Bridgestone sites on the web, including one at this link.