I don't know if it helps, but a common - and not unsimilar - phrase (or verse) used in such situations is
Heile, heile Segen
drei Tage Regen
drei Tage Sonnenschein
's wird bald wieder besser sein.
Note that this is positively used in South and South-West Germany, so covering the area you mentioned.
Some ideas contributed in comments
This answer (or non-answer) received valuable feedback from commenters, partly with new creative attempts to explain the wording. I summarize and comment the thoughts refering to the enigmatic word "soli":
1) Soli = Segen?
Marzipanherz asked: Is there a south/southwest German form of Segen that sounds similar to Soli? My answer: I do not see a lexical relation between Segen and soli in German or a German dialect.
2) Soli = solacium?
One thing I was pondering is if soli here could be derived from Latin solacium (= Trost = comfort, solace), as Latin was broadly used in catholic churches up to the 1960s. But that's a very far-fetched idea. Villagers did not usually integrate Latin terms they heard in church into their everyday vocabulary. Famous exceptions are sacramentum and crucifixum which both are used as curse words (Sakrament! Kruzifix! Zefix!). But I have never met such a Latin church derivative in nursing rhymes.
3) Soli = sodele?
Beta said: Could soli be a diminutive form of "so" (as in södele or sodele), which would fit perfectly in a comforting situation. My answer: I am not sure about this idea. Having grown up in the Swabian South-West (with part of my family being farmers and active users of dialect) I first thought: no, seeing "soli" as representation of "sodele" does not make sense. On second thought, however, one could indeed imagine that there once was a verse like "Sodele, sodele, sodele" which got distorted over time by users of the language detached from the area (emigrant-settlers to Eastern Europe in 18th century?).
Or perhaps the grandmother said actually "sodele", but Emma (who asked) just wrote it down as "soli" because she is not aware of a word "sodele"?
4) Soli = soll ich?
Somebody suggested that soli could stand for soll ich (shall I / should I) . My opinion: the words taken isolated, yes, this makes sense. Written soli would be pronounced quite similar to what speakers of South-West German dialects say for "soll ich". However, I struggle to fit "soll ich" into the context of the nursery rhyme.
PS: A short search with one of our preferred look-stuff-up machines brought this variant of the same nursery rhyme:
Heile, heile Segen
sieben Tage Regen,
sieben Tage Sonnenschein,
wird alles wieder heile sein.
Heile, heile Segen,
sieben Tage Regen,
sieben Tage Schnee,
tut dem Kind schon nicht mehr weh.