One idea depends on Alfa Romeo remaining in F1 for a few more years as a constructor until 1990 (instead of 1985) and from around 1986 producing an alternate successful 1.5 Alfa Romeo 415T* 4-cylinder twin-turbo engine that can be supplied to Minardi prior to being replaced by the 3.5 Alfa Romeo V10* from 1989.
Another would be Minardi managing to become somewhat regular fluke point scorers, with the highlight being Pierluigi Martini somehow managing to beat Ayrton Senna to win Minardi’s first and only GP at the 1991 Portuguese Grand Prix.
From there Minardi would need a well paying sponsor such as Red Bull like HesitatedEye mentioned, which would allow Minardi to acquire, rename and push Faenza Calcio football team to Serie A. It would interesting for both the ATL F1 and Serie A Minardi teams managing to get Minardi merchandise to be very popular and grow into a well-known brand.
Apparently OTL Minardi originally to receive the 1.5 Alfa Romeo V8 Turbo for its debut season in 1985, only for Alfa Romeo to withdraw refusing to supply the engines and forcing Minardi to use Motori Moderni's 1.5 V6 Turbo. -
https://www.unracedf1.com/gian-carlo-minardi-interview/
*- For OTL Alfa Romeo 415T F1 engine see Ligier SJ29
*- Known as the Alfa Romeo Tipo 1035 / V1035, the OTL V10 was originally intended for F1 once turbos were banned with the V10 later used in the Alfa Romeo 164 ProCar and at one point in the Group C Alfa Romeo SE 048SP.