Liu Yu followed the precedents set by the Three Kingdoms and Jin and which would be followed by many subsequent dynasties until the Yuan- he named his new regime after the noble title he had held under the previous one. In 418, he had been granted a fief in the northeast of Jin territory and elevated to the rank of Duke (公 Gong). Because this fief was near the territory of the ancient Song state, his title became "Duke of Song." Cao Pi was "King of Wei" (魏王 Wei Wang) under the Han before he founded the Wei dynasty; Sima Yan had been "King of Jin" (晋王 Jin Wang) under Wei before he established the Jin dynasty. Later, Li Yuan would establish the Tang dynasty because he'd held the title "Duke of Tang" (唐国公 Tang Guo Gong) under the Sui. Although the custom of overthrowing a state from within and naming it for a fief technically originated with Cao Wei, it could be argued that this is more or less consistent with the practice of Ying Zheng and Liu Bang, who named their empires for the independent kingdoms they had held before unification.

As to why Liu Yu chose this "feudal" name rather than trying to revive Han as Liu Bei and Liu Yuan did- unfortunately, the main source for the Liu Song state, the Book of Song, is silent on this matter. It may be that enough time had passed that the Han name had lost some of its cachet and adherence to precedent mattered more. It may also have something to do with the fact that Liu Yu was not descended from Liu Bang (as Liu Xiu and Liu Bei were), but from one of his brothers. Or the poor fortunes of Liu Bei's Shu Han and the Xiongnu Northern Han (which quickly dropped both its Han name and its pretensions to imperial lineage after about a decade anyway) may also have steered Liu Yu away. In any case, we have no source which explicitly states his thinking on this matter.