Nice to meet you and welcome to the community!!
Regarding the topic, I find Option 3 to be the most plausible for reasons I'll explain below. I'll be covering each option and giving my thoughts about how plausible they are in terms of the story. I'll also give some of my theories so bear with me. That said, here we go:
1. The mothers being friends might be possible since most of the story revolves around Tootsuki and the idea that Jou's whole philosophy about cooking and love is hinted to have come from Gin and Soma's mom. However, I find the accident story to be a bit unrealistic and unrelated to Jou never graduating and Azami's obsession with gourmet cooking. After all, based on Soma's and Erina's ages (15/16 & 16 respectively) Jou would have been 23 and Azami 21, long past graduation. About Azami's obsession, we'll need more info but I feel it's from his upbringing rather than any sudden shock.
2. I find this highly unlikely as it would really break the fandom in half which isn't needed right now. Plus, the author seems to be developing the SomaXErina ship so having them be half- siblings would really throw a wrench in that. Also, I find it unrelated to Jou not graduating and Azami's obsession for same reasons as above.
3. This seems the most likely as so far most assumptions of them being related is based on both of them having attended Tootsuki with Jou and Azami with Erina’s mom being the more likely since it seems that Azami married into the family. However, Jou could just as easily have met Soma’s mom after he left Tootsuki (maybe he went into their shop and had a Shokugeki with her and so on…). At this point, Erina’s mom seems to have passed away when Erina was young, since otherwise I can’t find myself to believe that she would have not known or done anything about Azami’s brainwashing if she was alive unless she was even worse than Azami that she was banished like him.
I personally feel that the two mothers are not related to each other (as far as the manga suggests). However, Jou might have known Erina’s mom at Tootsuki since she was a Nakiri and so presumably learnt cooking there as well. I feel that Jou met Soma’s mom at a restaurant she was working in (maybe the Yukihira family shop?) and that their relationship might have started with a Shokugeki or tasting challenge (since Jou’s advice to Soma about cooking/love was to find someone that would make him give all the food he made to her), eventually culminating in their marriage and Soma being born.
So that’s all from me! What do you think?