Is there any biblical justification for a monarchy in either the New or Old Testament of the bible (ie a system of government with a monarch at the head) and if so where in the bible do we see this?
In order to establish if there is any biblical justification for a monarchy, we need to go back to the Old Testament to the time when God raised up judges to govern the Israelites after they took possession of the Promised Land. God knew a time would come when they would want to be ruled by a human king, just like the nations that surrounded them. God both predicted and permitted this saying:
When you enter the land the Lord your God is giving you and have taken possession of it and settled in it, and you say, ‘Let us set a king over us like all the nations around us,’ be sure to appoint over you a king the Lord your God chooses” (Deuteronomy 17:14).
In the period that preceded the monarchy, Israel had no king; everyone did as he saw fit (Judges 21:25). God raised up Samuel to lead the people (1 Samuel 3:4). All of Israel knew that Samuel was established to be a prophet of the Lord (1 Samuel 3:20). Samuel judged Israel all the days of his life, and when he was old he made his sons judges over Israel (1 Samuel 8:1). Israel rejected the sons, refused to obey Samuel, and demanded a king (1 Samuel 8:19–20). When Samuel reported their request to God, the Lord answered, “Listen to them and give them a king” (1 Samuel 8:22).
Saul was the first anointed king but after he deliberately disobeyed God he reigned unsuccessfully from 1049 BC to 1009 BC when David replaced him (2 Samuel 2:4). Solomon became king in 971 BC, possibly two years before his father David died, and reigned until 931 BC. After the death of Solomon, the kingdom was divided. Ten tribes formed the Northern Kingdom, called Israel; Judah and Benjamin formed the Southern Kingdom, called Judah. The date of the division of the kingdom is approximately 931 BC. https://www.gotquestions.org/kings-Israel-Judah.html
In the New Testament, Romans 13:1-5 establishes the principle that the authorities that exist (whether monarchs, emperors or dictators) are God’s servants, with the power to punish wrongdoers.
Even as far back as the time of Jacob, it was prophesied that the sceptre would not depart from Judah, nor the ruler’s staff from between his feet until the coming of the one “to whom it belongs” (Genesis 49:10). This is taken as a Messianic prophecy, fulfilled initially in David, and ultimately in Christ. Zedekiah, the last king of Judah was told to remove the crown from his head (Ezekiel 21:26), showing who is really in charge.
Queen Elizabeth II, who died on Thursday 8 September, age 96, acknowledged that she was a servant queen who served not only her people but also the King of Kings, and Lord of Lords. In 2002 she said:
I know just how much I rely on my faith to guide me through the good times and the bad. Each day is a new beginning. I know that the only way to live my life is to try to do what is right, to take the long view, to give of my best in all that the day brigs, and to put my trust in God ... I draw strength from the message of hope in the Christian gospel. Source: The Servant Queen and the King she serves – published by the Bible Society (Christmas 2002)
Is there a biblical basis for having a monarchy? Yes. God permits Kings and Queens to rule, but as with all in authority, they will ultimately answer to God for their actions. A time will come when God does away with human rulers and Christ Jesus will be established as the King of His Kingdom, and every knee will bow before him (Isaiah 46:23; Romans 14:11; Philippians 2:10).
In this age, God does not ordain any particular type of government. Rather, He reveals the purpose for any government of this age. And, accomplishes that purpose through many forms of government until a monarchy is established in the next age.
The biblical justification for a monarchy has two aspects:
What is allowed as government now in this fallen age.
What will be the government in the next age.
"For there is no authority except from God, and the authorities that exist are appointed by God.", Rom 13:1
The various forms of government, from monarchy to republic to worse, are all under sovereign control of Jesus Christ.
"He removes kings and raises up kings", Dan 2:21
The Book of Daniel is God's revelation for man's government unto the end of this age. In short, it means destruction is decreed unto the end of the age. No form of man's government is going to work due to man's fallen nature. Much of man's government are described as beasts by God per how they operate. They are dumb brutes, predatory and destructive in nature. They're all terrible, even the best ones. But, God is all wise and allows the destruction for a greater cause,
"Nevertheless they will be his servants, that they may distinguish My service from the service of the kingdoms of the nations.", 2 Chrn 12:8
It's not the form of government that's key, it's their purpose:
"From one man he made all the nations, that they should inhabit the whole earth; and he marked out their appointed times in history and the boundaries of their lands. God did this so that they would seek him and perhaps reach out for him and find him", Acts 17:27
What little easy prosperity and leisure has come in this age results in rapid deterioration, tyranny, persecution, war, etc. again, due to man's fallen nature. God allows it, so that people might seek Him. And, it is in persecution His Church grows and personal transformation takes place. Wisdom indeed. God uses His enemies and man's sinful nature to accomplish His purpose.
Ultimately, this age's government is going to result in this,
"when the power of the holy people has been completely shattered, all these things shall be finished.", Dan 12:7
"And the government will be upon His shoulder.", Isa 9:6
while all governments of this age are destroyed,
"it shall break in pieces and consume all these kingdoms, and it shall stand forever.", Dan 2:44
For now, the Holy Spirit restrains government for the purpose of growing the saints.
"And now you know what is restraining, that he may be revealed in his own time.", 2 Thess 2:6
Without that restraint, the world would rapidly lead to unprecedented chaos and tribulation, which it will once that restraint is removed,
"For there will be greater anguish than at any time since the world began", Matt 24:21
This situation will be be cut off, though...
"The kingdoms of this world have become the kingdoms of our Lord and of His Christ, and He shall reign forever and ever!", Rev 11:15
Christ will reign as King of Kings, Lord of Lords for 1,000 years on the earth.
His saints will rule with a rod of iron over the nations, e.g. Ps 149. They rule, because they willfully govern by His Will, not their own. And, that by virtue of being made in His Image - which is His nature.
This is the monarchy intended for all mankind going forward.