"Da solo" is closer to "on one's own", "solo" is alone; both would work, just like in English, although "da solo" is better because it implies independence rather than loneliness.
Or "by herself"--it seems that both "on one's own" and "by one's self" can reflect the self-sufficiency meaning, whereas "alone" (or even stronger, "all alone") emphasizes the loneliness aspect.
Perhaps the da is to help distinguish solo/a/i/e (an adjectival phrase) from the adverb solo, which can mean "only", e.g.,:
vive solo per praticare il violino = "He only lives to practice the violin"
vive da solo per praticare il violino = "He lives alone to practice the violin"
In the woods. Now i must bring her a basket of food while wearing my bright red cape.