Chinese characters haven’t undergone that much change since the Qin dynasty (est. 221BC) but there were a couple rounds of ‘modernisation’ or simplification in the 20th century. The mistranslation into English of this shop is the unfortunate consequence of the simplification of two Chinese characters.
As seen here the character on the left, 发, has TWO complex or traditional forms:發 and 髮. The first, 發 fa1, means ‘issue, send out, utter, happen’ and when combined with another character 展zhan means ‘develop’. The second, 髮 fa4, is related semantically as well as phonetically and means ‘hair’, or something that shoots out from the body. Perhaps because of this semantic and phonetic similarity, these two characters were merged into one: 发.
This merging of two characters is perhaps not explained in the run-of-the-mill Chinese-English dictionary, thus leading to the strange name of this salon. The salon owner wanted 发 to mean 髮 ‘hair’ and not 發 ‘develop’. So, the name of the shop should be ‘Hair of Road’? Not quite.
There’s still the matter of the second and third characters. The second character, 之， does mean ‘of’ (in modern Chinese), but typically X 之 Y is translated as ‘the Y of X’. And the third character, 道 dao4, does often mean road or way, but also means Dao or Tao, as in Taoism.
Finally, translating 发 as hair and 道 as Tao, then putting them in the right order leads us to what was probably the owner’s original intent: The Tao of Hair; what I’m sure everyone would agree is a much better name for a salon.