I think that "το δικό μου" refers to "όνειρο". It's common in greek poetic language to say that something "στοιχειώνει το όνειρο".

There are just 3 lines before (and that's all):

Απλώνει τα πέπλα της/ η νύχτα, ντύνοντας με στοργή/ τ'όνειρα του κόσμου.

Could Το δικό μου refer to όνειρο?

I suppose it is taken from a poem or song, so there must be a reference in an other verse. Most possibly it means thought "my mind" or "my head" or so. :)