In the operetta Die Csárdásfürstin they sing in the part 2 about "Mädis vom Chantant".
While I do understand Mädis stands obviously for Mädels (or are there additional connotations?), I fail to research successfully which source this variant has (some local dialect?)
"Mädi" is certainly an abbrevation of "Mädchen" and is used as an affectionate form, although it seems to be outdated nowadays.
It can also be a short form of "Magdalena" or "Margarete". Here are some examples: 1, 2, 3, 4, 5, 6. These examples refer mainly to elderly women which indicates that this use of "Mädi" could also be be outdated meanwhile. Interestingly "Mädi" seems to be a popular name for female animals. See 1, 2, 3, 4, 5.
Clearly in the operetta Die Csárdásfürstin the word "Mädis" stands for "Mädchen" or "Mädels", i.e. young women. It was a commonly used form, especially in Austria. Let us look at some other occurences of "Mädi".
The operetta Mädi by the Autrian composer and songwriter Robert Stolz. See here and here ("Mädi, mein kleines Mädi").
The song Ich denk' an Mädi die ganze Nacht by the Austrian-born composer Walter Jurmann.
The song Mein Liebling heißt Mädi which was published in the Wiener Bohème-Verlag. See here and here. It is the German version of Linger awhile and the German text was written by the Austrian librettist Fritz Löhner-Beda who was murdered in Auschwitz.
The song Wenn mein kleines Mädi weint by the Austrian composer Heinrich Strecker. See here and here.
The children's book Professors Zwillinge - Bubi und Mädi by Else Ury. See here. The author was German (born in Berlin) and murdered in Auschwitz.
The full libretto to Csardasfürstin by Emmerich Kálmán can be found here, but offers little.
Literature hits restrict to "Wie Uli der Knecht glücklich wird" (1841), by Jeremias Gotthelf, pseudonym of a Swiss author:
Zornig war Vreneli aufgesprungen im Wägeli: »Mädi, willst du den Hut geben oder nicht ? Was braucht Uli einen Meien ? Sei mir nicht ds Hergetts, einen Meienstock anzurühren!« Als Mädi nicht hören wollte, ...
as well as "Märchen" (1907) by Peter Altenberg, an Austrian author:
Irgendwo sagte eine Kinderfrau zu einem Kinde: »Na warte, du schlimmes Mädi, ich werde es dem Gärtner sagen – – –«. [Altenberg: Märchen des Lebens. Deutsche Literatur von Lessing bis Kafka, S. 1171 (vgl. Altenberg-Märchen, S. 103)]
The first example also exhibits a concentration of diminuitives quite typical in a dialect.
I consider the word as being used too seldom for a well-established connotation, but especially the pejorative meaning which the counterpiece Bubi sometimes carries, seems to be lacking in Mädi.
In addition to @guidot's answer:
Nicknames/petnames ending on -i are a common thing in German, especially in the south, as well as just taking names and making up a short form that ends on -i. Here obviously they're doing just that with "Mädchen". It's not much different to building a noun like "girly" from "girl" in English. Not really original, and could have happened at any time.
Adelheid => Heidi
Susanne => Susi
Eckart => Ecki
Manfred => Manni
Gespons (old word for spouse) => Gspusi
Mädchen => Mädi
I haven't heard "Mädi" much, and I don't think it has widely known connotations, especially it's not a (widely known) word for prostitutes (which the "Mädis vom Chantant" seem to be) or dancers or anything beyond what can be told from them being called girls. It may have had connotations back when the libretto was written, but I don't know the word from other texts from that period, and when searching Google for "die Mädis" the first pages of what comes up are "Die Csárdásfürstin" references.
