If I were to try to disuade a Jew from suing another Jew in court, without first going to a beis din, what Biblical and rabbinic sources should I cite as proof?
There's a whole article by Rabbi Yaacov Feit about this in the Journal of Beth Din of America, vol. I called, unsurprisingly, "The Prohibition Against Going to Secular Courts." It begins:
The Torah states (Exodus 21:1), “Ve’eleh ha-mishpatim asher tasim lifneihem,” “And these are the statutes which you shall place before them.” The Talmud (Gittin 88b), sensitive to the word “lifneihem”, deduces “lifneihem- velo lifnei akum,” “Before them- but not before gentiles.” As such, the Talmud understands that there is a prohibition against bringing disputes to be adjudicated before gentile courts.
Now practically -- if someone doesn't want to use a Beis Din, it's unlikely that you can show them a journal article or a Gemara and "aha!" that's going to change their mind. You can, however, point out that a beis din is faster, cheaper, and quieter than going to courts. But you know what else is? Arbitration or mediation. And (as the article points out), those are allowable as well. The only halachic prohibition is going to a different system of laws. So maybe you can have them consider a professional arbitrator or mediator. (They don't have to be Jewish.)