"The Conscience of the Court" is a short story originally published in The Saturday Evening Post on March 18, 1950. It is set in Jacksonville, Florida, at the trial of fictional character named Laura Lee Kimble, an uneducated black woman from Savannah, Georgia, now living with her (temporarily absent) employer in Jacksonville. Here is the relevant excerpt:
"Charged with felonious and aggravated assault. Mayhem. Premeditated attempted murder on the person of one Clement Beasley. Obscene and abusive language. Laura Lee Kimble, how do you plead?"
Laura Lee was so fascinated by the long-named things that they were accusing her of that she stood there tasting over the words. Lawdy me! she mused inside herself. Look like I done every crime excepting habeas corpus and stealing a mule.
Harold Wentworth, American Dialect Dictionary (1944) offers the following relevant information:
Law!, La!, Lawzy! Also Lawdy!, Law sakes!. Law me!, &c. Cf. land 2 [which reads "As or in exclam.: Land!, Land's sake! Landy! &c. 1834–1918 Land's sake(s), land sake(s)."] ... 1925 s[outh]w[estern] S[outh] C[arolina]–e[astern] G[eorgi]a Svannah R[iver] Negro. Everyday speech. 'Lawdy Jedus, Ah's so fraid.' ... 1938 Fl[orid]a–G[eorgi]a Suwannee R[iver] Lawsy! ...
As Wentworth's examples suggest, "Lawdy me!" is a dialect pronunciation of "Lordy me," which (according to the OED) is itself a contracted form of "Lord help me." As an exclamation, however, it isn't any more specific than "Oh me!" or "Oh my goodness!" or "My, my, my!" or "Good grief!" The expression can be used across a wide range of situations, in forms ranging from a cry of despair uttered in the midst of a dreadful crisis to a sort of quiet, sing-song musing.
A famous U.S. folk song called "Reuben's Train" (memorably performed here by the Holy Modal Rounders) uses the refrain "Oh me, Oh Lordy my!" after each piece of narrative action in the song:
Oh, Reuben had a train, and he put it on the track
And he run it to the Lord knows where
[Chorus] Oh me, oh Lordy my, run it to the Lord knows where
What did the hobo say to the bum? Got some cornbread save me some
I b'lieve, praise God, I'm comin' down
[Chorus] Oh me, oh Lordy my, I b'lieve praise God, I'm comin' down
[and so on...]
Another traditional song—recorded many years ago by Alan Lomax as Georgia penitentiary inmates sang it as a convict-crew work song—is titled simply "Oh Lawdy Me, Oh Lawdy My." Though many of the lines of that song begin with words such as "Lord, I'm goin' where I've never been before," it isn't a gospel song. In fact, it isn't particularly religious at all, except to the extent that formally calling the Lord to witness your troubles retains a theological dimension even when the usage is habitual and not prayerful.
In Hurston's short story, the crucial clue to how the character is using "Lawdy me" is in the description "she mused inside herself." The character isn't using the phase in the as if it were a fervent prayer or as an expression of fear and desperation, but in a thoughtful, unhurried way.