General Rules of Thumb
There really isn't any single firm rule of thumb.
But, there are rules of thumb that apply in particular circumstances that I know from experience practicing in these areas. Legal claims are highly heterogenous and unless you know more about the type of claim and how strong it is, you can't value it. Some claim type specific rules of thumb are set forth below. Some kinds of claims settle at a high percentage of the claimed value (e.g. debt collection cases with solvent defendants), while others settle at a low fraction of it (e.g. weak securities fraud lawsuits).
The amount of the settlement depends upon (1) how solid the claim is, in fact, (2) how much it is likely to cost to litigate, (3) whether there is a likelihood that a prevailing party will recover attorneys' fees, (4) the ability of a defendant to pay a judgment, (5) the ability of the plaintiff to persevere and to finance a litigation to its completion, (6) the existence of plausible counterclaims, (7) the ability of a plaintiff to pay counterclaims or sanctions that are awarded, and (8) the long term strategic impact of a settlement or award in future cases or disputes.
If a claim has only negligible prospects of success and does not have a reputational cost if settled, a "nuisance settlement" is considered and is usually some fraction of the estimated cost to the defendant of litigating the case to the point at which it is dismissed (usually in pre-trial motion practice).
If a claim has a material chance of success, the rule of thumb is that it will settle for the probability of success times the most likely damages award if it is successful. But, this benchmark adjusted for litigation costs and for the fact that, in practice, while cases where damages are disputed often settle, cases in which there is a serious doubt about the existence of any liability very frequently do not settle.
More generally, cases tend to settle "in the shadow" of what is likely to happen if the case is litigated and goes to trial, with an adjustment for the costs associated with that process and for the uncertainties involved.
For example, medical professionals very rarely settle any claim that is not completely open and shut in terms of liability and a large damages award, because a settlement payment must be publicly reported and can harm the professional's reputation and long term economic prospects.
In debt collection cases where there is a more or less undisputed or just moderately quibbled with debt due, the primary factor in setting a settlement amount is the ability of the debtor to pay and the amount that could be recovered if the debtor filed for bankruptcy rather than litigating the case on the merits. This, in turn, can hinge both on the assets of the debt, and the priority that a debt would have in a bankruptcy.
In tax cases where there is not a serious dispute about the amount actually owed, a waiver of penalties, but not of interest or any of the principal amount of the tax owed is common. But, there can be significant reductions if liability for the tax is credibly disputed or if the ability of the taxpayer to pay the tax debt is seriously in question.
In personal injury cases and other tort cases, the amount of available insurance coverage is frequently an anchor around which settlement discussions hover in a strong liability case, unless the damages incurred are obviously far less than the available insurance policy limited.
In employment cases where an employee at will has been wrongfully terminated, there is a rule of thumb that that six months pay is a common fair settlement figure in a typical case, before adjusting up or down for likelihood of success on liability if the case is stronger or weaker than average.
In case where a plaintiff is entitled to both economic damages and non-economic or punitive damages if the plaintiff prevails, the estimated non-economic and punitive damages are typically expected to be on the same order of magnitude as the economic damages unless state law limits non-economic or punitive damages more severely. Also, economic damages that aren't very solidly determinable are usually heavily discounted or ignored.
But, if liability is exceptionally clear (e.g. a tortfeaser is caught red handed on videotape or in an incriminating document), the value of non-economic or punitive damages increases dramatically.
There is an academic literature on the subject in tort cases, especially medical malpractice cases, where insurance companies can make large data sets available, and to a lesser extent in other kinds of cases.
But, people who are actually negotiating settlements are more likely to make reference to reports of jury verdicts in similar cases (which there are services that report), rather than to settlements in comparable cases.
Good statistical information is hard to gather in most circumstances, because, as noted in a comment to another answer, the amount of most settlement agreements is confidential. One exception is settlements with government entities which are public since they must be disclosed as part of the governmental budget process.
Another technical issue is that many states prohibit an "ad damnum" clause making a demand for a specific sum due in a complaint initiating a lawsuit.