Even if a great number of commentators opt for a ‘demonic’ explanation, I think there is no need – in this case – to involve demons in play.
(I) First of all, please, note the different phraseology utilized in the passage, focusing on the expression “(a) spirit from (God)” (1 Sam 16:14).
In the first instance we have a structure about a property-related concept (without any prefixed particle), namely
(‘a spirit belonging to יהוה ‘ > ‘a spirit of יהוה ‘).
In the second instance, we have the following structure (with a prefixed particle), namely
רוח־רעה מאת יהוה.
It is clear that (not taking into account the adjective רעה, ‘evil’) – also if you don’t read Hebrew at all – the phraseology of the two expressions is very different. The part of speech that makes a difference is the -M prefixed to the AT particle (I have underscored the -M prefixed). Just now we can conclude that – for logic - we are not in the presence of the same meaning, as if both expressions would suggest ‘(a) spirit from (God)’ > ‘(a) spirit of (God)’. A different expression’s construction must correspond to a different meaning. Moreover, if the second instance were to enhance the ownership of God over the ‘spirit’, the text were רוח־רעה יהוה, without the term מאת [MAT]. Then, this term (מאת [MAT]) rules out the possibility here the Bible refers to ‘a spirit belonging to יהוה ‘ > ‘a spirit of יהוה ‘. This inserted particle, instead, must indicate slight, but important, shifting of meaning (we will see it later).
(II) If we were in the presence of a diabolical possession, as a lot of commentators sustain, in what manner a ‘harp, lyre’ [NGN] was able to heal Saul (“[…] and you will be well”, 1 Sam 16:16, ESV)? Was a sound of a harp (or, lyre) enough effective to impede a demon to perform his evil deeds through a possessed (by him) man (in this case, Saul)?
For me is hard to imagine the answer of Jesus to his apostles that asked (really) him ‘Why could we not cast it [the demon] out?’ (Mat 17:19, ESV) - along with the answer (imaginary) of their Lord - ‘How clever of you! You’ve forget your harps at your houses!’.
Not to mention the occasion when the lyre/harp’s sound didn’t work on Saul (1 Sam 18:10)…
(III) An effect of the ‘bad spirit’ was not the God’s-inspired prophesying of Saul (1 Sam 18:10), like someone say. In fact, see how these Bibles translated the expression, along with a comment (bold is mine):
“he [Saul] went into a frenzy within the house”, Robert Alter
“he [Saul] raved within his house”, ESV
“and he [Saul] raged in his house”, NAB
“and he [Saul] fell into a frenzy while he was indoors”, NJB
In a similar way translate NLT, NRSV.
“He prophesied in the midst of the house, that is, he had the gestures and motions of a prophet, and humoured the thing well enough to decoy David into a snare, and that he might be fearless of any danger and off his guard; and perhaps designing, if he could but kill him, to impute it to a divine impulse and to charge it upon the spirit of prophecy with which he seemed to be animated: but really it was a hellish fury that actuated him.” (Matthew Henry, Commentary on the Whole Bible, ad locum)
At this point, before I present an alternative explanation, we will take a look to the personality of Saul, as far as the Bible presents it to us.
After a Samuel’s praise to the Saul’s family:
“And Saul answered and said: ‘Am not I a Benjamite, of the smallest of the tribes of Israel? and my family the least of all the families of the tribe of Benjamin? wherefore then speakest thou to me after this manner?’” (1 Sam 9:20-21, JPS)
Later, at Mizpah:
“And Samuel bringeth near the whole tribes of Israel, and the tribe of Benjamin is captured, and he bringeth near the tribe of Benjamin by its families, and the family of Matri is captured, and Saul son of Kish is captured, and they seek him, and he hath not been found. And they ask again at Jehovah, ‘Hath the man yet come hither?’ and Jehovah saith, ‘Lo, he hath been hidden near the vessels’” (1 Sam 10:20-22, YLT).
Some comments (bold is mine) to the ‘hiding’ of Saul:
“For he [Saul] knew that God had chosen him; and Samuel had anointed him already. He did it therefore simply from humility and modesty. ‘In order that he might not appear to have either the hope or desire for anything of the kind, he preferred to be absent when the lots were cast’ (Seb. Schmidt).” (Keil & Delitzsch Commentary).
“he [Saul] hath hid himself among the stuff — Among the carriages or baggage of the people there assembled. This he probably did from a sense of his own unworthiness.” (Joseph Benson)
Compare, also, the reaction (or the absence of it) of Saul in the subsequent occasion described in 1 Sam 10:17-27.
So, the Bible introduces Saul as a modest, also a shy person.
You have surely acquainted with shy, bashful people. Unhappy, many of them remain in this condition for all their life. Other reach – happily - a balanced view of themselves. But, in some cases, they succeed somehow to loosen their psychological block, but developing at the same time – gradually – a tendency to go to the opposite extremes, maybe becoming loquacious people, later being self-centered, until they may reach also a condition of a permanent boastfulness and pride.
It seems to me this last description is apt to Saul later mental condition. In a short time, Saul swung from a bashful hiding of himself (1 Sam 10:22) to a building of a monument for himself (1 Sam 15:12). Moreover, his subsequent reaction to the women chants in David’s praise confirms he became a pride (1 Sam 15:22-23), also an envious man (1 Sam 18:7-9), at all.
Can all these information help us to understand the sense of the passage at issue?
Instead to hypothesize that the prefixed -M in this passage points exclusively to an ownership of God over the ‘spirit’ (or, alternatively, a ‘provenience’ by God) we have to remember that this prefixed particle has also a different nuance of meaning.
Some passages (bold is mine)
“And Moses spoke so to the children of Israel: but they hearkened not to Moses, by reason of anguish [מקצר] of spirit, and cruel bondage.” (Exo 6:9, Webster)
“Yahweh loved you and chose you not because of your great number [מרבכם] exceeding all other peoples, for you are fewer than all of the peoples” (Deu 7:7, Lexham English Bible)
“Because of the voice [מקול] of him who says sharp and bitter words; because of the hater and him who is the instrument of punishment.” (Psa 44:17, Bible in Basic English)
“because of the voice [מקול] of the enemy, and because of the oppression of the sinner: for they brought iniquity against me, and were wrathfully angry with me” (Psa 55:4, Brenton).
“He was wounded and crushed because of our sins [מפשענו]; by taking our punishment, he made us completely well.” (Isa 53:5, CEV).
Similarly in Aramaic (see Dan 5:19).
From these examples we now understand that M- possesses also a different nuance of meaning. Here are some reference work.
Wilhelm Gesenius, in his A Hebrew and English Lexicon of the Old Testament pointed all the nuances of meaning of M- (as derivative from MN < MNN). One of these is described on page 582 (‘f’): “Of the ‘remote cause’, the ‘ground’ or motive ‘on account of, because of’, ‘for’ which any things is done. Joel 4,19 [3,19] [...] ‘for the violence to the children of Judah’. Is[aiah] 53,5 […] ‘for (on account of) our transgressions. Pro. 20,4. Deut. 7,7. Josh. 22,24. Cant. 3,8 Zech. 8,10. […]”
Similarly, also The Dictionary of Classical Hebrew (Sheffield Academic Press, 1993), on the pages 342-343, under the sub-entry ‘8. of cause, on account of, because of, for (reason) of, through, at, e.g.’; Parkhurst, ‘because of, by reason of’; and Davidson, “[…] (c) because, on account of […].”
So, the more correct translation of the passage in issue (1 Sam 16:14) could be:
“A flux (or, 'a spirit') of יהוה departed from Saul, but terrified him an unsuitable flux (or, 'a spirit'), on account of יהוה“.
Knowing God had departed from him, Saul developed in his mind a very strong mental pain. The full awareness that God abandoned him, once and for all, was driving him mad. He made himself ill about the fact he lose his former approved position before God.
Interestingly, once I described the Saul’s psychological profile according to the Bible to a psychoanalyst (avoiding to tell him who is the man [Saul], and the source of information of mine [the Bible]). So, he told me that this mental disorder is named ‘obsessional neurosis’.
I hope these information will be useful for you.