The reason that the translations are all over the map is:
Some Hebrew manuscripts connect the first word of the following verse with the last word of Psalm 42:5, yielding the phrase "My ever-present help, my God"
The meaning of the Hebrew of Psalm 42:11 is uncertain.
The above is pointed out in the apparatus of the JPS Tanakh in the Oxford Jewish Study Bible (2nd ed.).
It might be worthwhile to appeal to the Greek Septuagint in this case. Although a translation, even Jewish scholars agree that the underlying Hebrew text is several centuries older than the one consulted by the Masoretes. In the Septuagint, Psalm 42:5 (41:5 LXX) is (Brenton translation):
Wherefore art thou very sad, O my soul?
And wherefore dost thou trouble me?
Hope in God; for I will give thanks to him;
He is the salvation of my countenance.
A more modern rendering is in the Orthodox Study Bible:
Why are you so sad, O my soul?
And why do you trouble me?
Hope in God, for I will give thanks unto Him;
My God is the salvation of my countenance
The Holy Transfiguration Monastery Psalter (found here) uses similar language
Why art thou cast down, O my soul?
And why dost thou disquiet me?
Hope in God, for I will give thanks unto Him;
He is the salvation of my countenance, and my God
The phrase salvation of my countenance [σωτήριον τοῦ προσώπου] - soterion tou prosopou does not appear to be an idiom. "Soterion" (a neuter noun) appears to be an older form of the word used in the New Testament - "soteria" (a feminine noun) - and appears only in the Greek Old Testament, but only here in combination with prosopou. The Oxford Jewish Study Bible commentators do not indicate that it is some sort of Hebrew idiom.
Although soterion and soteria are usually translated to mean "salvation", they are also used to mean "healing" (in fact, the Greek Fathers seem to often discuss salvation in terms of a sort of spiritual healing). Augustine understands the phrase in this context:
My God is the saving health of my countenance.
My “health” (my salvation) cannot be from myself; this it is that I will say, that I will “confess.” It is my God that is the saving health of my countenance. For to account for his fears, in the midst of those things, which he now knows, having come after a sort to the “understanding” of them, he has been looking behind him again in anxiety, lest the enemy be stealing upon him: he cannot yet say, “I am made whole every whit.” For having but the first-fruits of the Spirit, we groan within ourselves; waiting for the adoption, to wit, the redemption of the body [Romans 8:23]. When that “health” (that salvation) is perfected in us, then shall we be living in the house of God for ever.