I think this will epend on the historical roots the name comes from. Socialism/Communism in its basic form is a very old concept and can arise from almost any cultural setting, so the name will depend on where it's coming from.
A version arising from radical Christian roots could be called Levellerism, Adamism (when Adam dalve and Eve span...), or Petrininsm (see Acts). Or, of course, Communism, since the term 'commune' was common in medieval Latin.
A more intellectually inclined version might come to be called something snobbish like Homoiopolitism (state of equals) or Omnipragmatism (property of all).
Collegialism or Sodalism lends itself to a less ambitious syndicalist conception.
And of course nearly every European country has some kind of 'good old tradition' when land was held in common and the village community functioned as one. IIRC it's Russkaya Pravda to the Slavophiles and was called teutsches Recht by the peasant rebels and their theorists and Saxon rights by Victorian intellectuals.