What's the etymology of sound' in 'sound in damages, tort, contract'?

One possibility is that legal use of sounding arose by working backward from the notion of a case's being "heard" by a court for purposes of resolving a particular claim. Here is the first part of the entry for hearing in Black's Law Dictionary, fourth edition (1968):

HEARING. Proceeding of relative formality, generally public, with definite issues of fact or of law to be tried, in which parties proceeded against have right to be heard, and is much the same as a trial and may terminate in final order. [Citations omitted.] Synonymous with trial, and includes reception of evidence and arguments thereon. [Citation omitted.] It is frequently used in a broader and more popular significance to describe whatever takes place before magistrates clothed with judicial functions and sitting without jury at any stage of the proceeding subsequent to its inception, and may include proceedings before an auditor. [Citations omitted.]

A hearing, understood broadly, is thus a proceeding at which any suit or claim is brought to a magistrate or judge for adjudication. It is no great stretch, then, to suppose that sounding is the aspect of a claim that, under law, merits a hearing by a magistrate. Here is the entry for "sounding in damages" in Black's Law Dictionary:

SOUNDING IN DAMAGES. When an action is brought, not for recovery of lands, goods, or sums of money, (as is the case in real or mixed actions or the personal action of debt or detinue,) but for damages only, as in covenant, trespass, etc., the action is said to be "sounding in damages." [Citations omitted.]

In other words, when a plaintiff brings a case to court for a hearing on the subject of damages, the plaintiff's action "sounds" in the hearing in relation to the subject that the law narrowly deems appropriate for that action. A suit for trespass, for example, would sound in damages (the appropriate compensation for the tort of trespass under law) but would not sound in replevin (recovery of lost goods) because a suit for trespass is not the appropriate action to file if the outcome desired is restoration of unlawfully taken property. The hearing is strictly attuned to the sounding that makes sense in connection with the type of compensation sought.

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A second interpretation of sounding arises from a broader notion of "wholeness." This sense of the sound as an adjective receives considerable attention in Noah Webster, A Dictionary of the English Language, volume 2 (1832), which bundles the legal definition of sound in the company of numerous other definitions related to being entire, unbroken, healthy, and nondefective:

SOUND, a. ... 1. Entire; unbroken; not shakey, split or defective; as sound timber. 2. Undecayed; whole; perfect, or not defective; as, sound fruit; a sound apple or melon. 3. Unbroken; not bruised or defective; not lacerated or decayed; as a sound limb. 4. Not carious; not decaying; as, a sound tooth. 5. Not broken or decayed; not defective; as, a sound ship. 6. Whole; entire; unhurt; unmutilated; as, a sound body. 7. Healthy; not diseased; not being in a morbid state; having all the organs complete and in perfect action; as, a sound body; sound health; a sound constitution; a sound man; a sound horse. 8. Founded in truth; firm; strong; valid; solid; that cannot be overthrown or refuted; as, sound reasoning; a sound argument; a sound objection; sound doctrine; sound principles. 9. Right; correct; well founded; free from error; orthodox. 2 Tim. i. "Let my heart be sound in thy statutes." Ps. cxix. 10. Heavy; laid on with force; as, sound strokes; a sound beating. 11. Founded in right and law; legal; valid; not defective; that cannot be overthrown; as, a sound title to land; sound justice. 12. Fast; profound; unbroken; undisturbed; as, sound sleep. 13. Perfect, as intellect; not broken or defective; not enfeebled by age or accident; not wild or wandering; not deranged; as, a sound mind; a sound understanding or reason.

Nevertheless, having made a case for the adjective sound as, in legal settings, being allied with notions of rightness and freedom from defects, Webster tucks an explanation of the expression "to sound in damages" under an entry for the verb sound in the sense of emitting audible noise or vocalization:

SOUND, v.i. [1.] To make a noise; to utter a voice; to make an impulse of the air that shall strike the organs of hearing with a particular effect. We say, an instrument sounds well or ill; it sounds shrill; the voice sounds harsh. "And first taught speaking trumpets how to sound." Dryden. 2. To exhibit by sound or likeness of sound. This relation sounds rather like a fiction than a truth. 3. To be conveyed in sound; to be spread or published. "From you sounded out the word of the Lord." 1 Thess. i. To sound in damages, in law, is when there is no specific value of property in demand to serve as a rule of damages, as in actions of tort or trespass, as distinguished from actions of debt &c. Ellsworth.

It thus appears that Webster, after all, finds the origin of "sounding in damages" not in notions of legal wholeness or rectitude, but in the emitting of something audible. In this respect, an interesting comment posted beneath this answer by EL&U site participant Steve may be pertinent:

I've always thought "sound" in this context is related to the engineering practice of striking something to test its integrity. Something "sounds in damages" means "if tested, will resonate with the remedy of damages". Cf. "Sound/unsound reasoning".

There are indeed several senses of sounding that involve testing the depth or validity of a thing being measured or assessed. But it remains to be demonstrated whether sounding in the legal sense arose from these operations.

It seems @SvenYargs missed a remote possibility:

Compare reconstructed Proto-Germanic *sundraz "separate, isolated, alone", reflected in English sunder, asunder, sundry, German sondieren, sondern, besonders, absondern., etc. Since our sounding did not make the list, I am less than convinced by my own idea, on phonetic grounds.

It makes sense semantically, at least. Not only are claims for restitution isolated from the main case (at least where the criminal/civil distinction holds, though these may be combined through adhesion), but after holding a claim for damages there might be an incentive to separate the claimed property from the losing party.

Legalese is known to conserve archaic features of language to the point that it becomes incomprehensible. Unless this turn of words was a recent creation, the similarity to modern meanings of sound may be coincidental, or formed by analogy if the original meaning had been all but forgotten.

In the same wane, he preposition in may hold additional information. Ger. Sühne "repenance, atonement" is cognate to sound "whole, healthy" (and Ger. gesund "healed, healthy", gesamt "summed, completed"), indeed, via *swono. In that sense, a connection to sound "sono, noise" at the root level is thinkable, as verbalized appology, ie. repanance. And if we seek and claim damages it is thinkable to repay in damages, though this is fickle.

It seems possible that the law-specific use of "sound" was imported from Law French. I see in The Law-French Dictionary Alphabetically Digested, Published by Nutt and Gosling, London, 1718, an entry for "Sourdant" ("-ant" being the French ending roughly equivalent to "-ing" in English), meaning "arising or springing out". Also "Soudra" = "They arose". (These are on pp 134 and 135).