The accordion is one of a number of what you might call Cinderella instruments in classical music. Its name derives from a Viennese patent of 1829, the development of free reed instruments having been stimulated by the introduction of a Chinese mouth organ, the sheng, into Europe in the late 18th century. The accordion was readily adopted into folk and popular music. Yet, although The New Grove traces the first "original composition of music of importance" for the solo accordion back to 1927, and a number of major figures have written for it, the repertoire has never achieved critical mass.
The uphill battle faced by the accordion can be gauged by the fate of the guitar, with us since the Renaissance and with a vast repertoire, but little of note by a composer of the first rank. In solo recitals guitarists rely heavily on arrangements and transcriptions, ironically, often of piano pieces written in imitation of the guitar.
The more recent mandolin, another member of the lute family, caught the attention of Vivaldi, it serenades lovers in opera, and was used by Mahler, Schoenberg, and Stravinsky. There's nothing, however, quite like the sound of the mandolin orchestras which can be found in a number of European countries.