I heard explained that it means that the LORD will still extend mercy in spite of sins. I have also heard that it means just the opposite-that an extended hand in ancient Near Eastern context means a hand administering punishment, or, at least, threatening or beginning to do so.
It is much better to determine the meaning of a phrase in the Bible from how that phrase is used elsewhere in the Bible rather than what scholars suppose was the custom in the ancient near east. To use non-Biblical sources to define the Bible would be a last and shaky resort.
The first usage of God stretching out His hand is in the Exodus (Exo 3:20; 7:5; 15:12) when God did many miracles to free His people from bondage. There are also many more verses there where Moses and Aaron as God's representatives stretched out their hands over Egypt to bring plagues.
For God to stretch out His hands is for Him to do something beyond the norm, namely miracles. Acts 4:30
In the NT, in Acts 4:30 God stretches out His hand to heal and do other miracles, but in the OT all, or at least almost all instances of God stretching out His hand are to do judgments.
Besides the Exodus examples there are 2Sam 24:16; 1Chr 21:16; Job 30:24; Ps 138:7; Jer 6:12; 15:6; 51:25; Ezek 6:14; 14:9,13; 16:27; 25:7,13,16; 35:3; Dan 11:42; Zeph 1:4; 2:13
Possible non-judgments of God's stretching out His hand are:
2Chr 6:32 - strangers will hear of God's stretched out arm and come to Israel.
Ps 136:12 - kindness to Israel, but judgment on Egypt, referring to the Exodus
144:7 - stretch out Your hands to rescue me. This could mean to enable me to escape, but looking at the verses before and after, it more likely means to smite those attacking me.
There is also Isa 65:2, I have stretched out My hands all the day unto a rebellious people ... .
This is a different Hebrew word for stretch. It actually means "spread out". This could refer to either judgment or mercy or both.
There are uses of stretching out the hands to help or to meet others where they are at, but the subject of these verses is not God: Prov 1:24; 31:20.
Back to the original verses in Isaiah referenced in the question, (to which I would add 23:11; 31:3): from the context these are all judgment. The meaning is that even though God has punished so much, He is still not done. "His anger is not turned away, but (or "and") His hand is stretched out still." In contrast to His anger being turned away, His hand is still stretched out. If it said, "His anger is finally turned away, and His hand is at last stretched out", then that would be an offer reconciliation.
See also Lam 2:8 He has not withdrawn His hand from destroying
The idiom of the "hand" in ancient Jewish thought was extremely significant. Hebrew is a very action oriented language and the "right hand" is the means of accomplishing most of that action. Thus it is used of God frequently.
- Ps 137:5 - If I forget you, Jerusalem, may my right hand forget its skill.
- Ps 118:5 - The LORD's right hand has done mighty things!
- Isa 41:10 - So do not fear, for I am with you; do not be dismayed, for I am your God. I will strengthen you and help you; I will uphold you with my righteous right hand.
- Isa 5:5:25, 9:12, 17, 21, 10:4 - the stretched out hand is a symbol of punishment against sinful nations. See also 2 Sam 24;16, Eze 16:27, 1 Kings 13:4, 1 Chron 21:16, Isa 14:26.
- Eze 2:9 - Then I looked, and I saw a hand stretched out to me. In it was a scroll.
- Isa 59:1, 50:2, Num 11:23 - the hand of the LORD is not to short to save.
Thus, the "hand of the LORD" is a means of protection for the righteous, punishment for the wicked, a source of good gifts, and a means of salvation.
That is, the "hand of the LORD" is depicted as a metaphor of God's action when it is stretched out, and in-action when it is "short" (not stretched out).
And His hand is still stretched out.
What does Isaiah mean by saying that the LORD's “hand is stretched out still”? (Is. 5:25;9:12,17,21;10:4)
Apostasy ,corruption, wickedness, and violence has permeated all levels of the Israeli society. The judges and officials use their power to plunder the poor , the lowly and the fatherless child, that come to them seeking justice and assistance.
God is patient and send his prophets , Hosea pleads: "Return, O Israel, to the Lord your God, For you have stumbled because of your iniquity'" Hosea 14:1 (NASB) but they do not listen, they refuse to change their ways.
In spite of all this, His (God's)anger does not turn away"and His hand is still stretched out" it is not a threat, and it is not a reconciliation offer, like you said ,this means that God has decided "to administer justice."