"Good records" is a bit of a vague standard, but I'm curious about what level of evidence survives from different times. For example, if there's an ancient Babylonian lawsuit where we have a survival of the basic verdict (shekels paid, etc), a Han-period Chinese lawsuit where we have some record of the attorneys, motions filed, etc, and a 16th century Venetian lawsuit that is the oldest for which a complete surviving transcript of the trial, then I'd be curious about all of them.
(I realize an "oldest x, biggest y, fastest z," type question are somewhat frowned on here, but I'm not sure I know enough about the history of litigation to ask about this without including some element of "throughout history," and searching for "earliest lawsuit" and such didn't get what I was looking for.
This is indeed a tough question to answer. I'll venture a try but I can't speak to Sumer, or to China, and hesitate to go into detail on Rome without my books on hand (I'm on vacation!). Nevertheless, some thoughts.
The English legal framework was well established by at least 1215. We know this because Magna Carta contains a provision that sttempted to create predictable venue options for private litigation. Magna Carta also originated the principle of habeas corpus, a frequent topic in criminal and security litigation. The Supreme Court, in Boumediene v. Bush, engaged in a thorough review of that concept's antiquity and development through the centuries. Other common threads in inter party disputes can be traced to the 1500s: in one example I'm fond of citing, a case reporter discussed how one case presented a unique issue of statutory interpretation -- that is, the very thing that modern courts deal with on a daily basis.
In this manner, then, good records trace the history of litigation in the Anglo-American tradition to at least 1215. Equally good (perhaps better) records demonstrate a rich legal tradition in the Roman Republic. A flaired Roman history might be more help for you on that issue. I encourage you, though, to look at Cicero's speeches in defense of his legal clients.