The word "raised" in Romans 9:17-18 and Exodus 9:16?

To help with this exegesis, you should consider Dan 4:25.

the most High rulers in the kingdom of men and gives it to whomsoever he will (Dan 4:25, KJV)

Here God makes a point of humbling Nebuchadnezzar, King of Babylon, to show that He chooses the leaders of the world and gives it to whoever He wills.

Given this case, why was that particular pharaoh in power during the time of God's deliverance of the Israelites? Was it because this pharaoh was a man after God's own heart, similar to David? No, quite the opposite! God already knew "the Pharaoh will not hearken unto you" (Exodus 7:4). He allowed the Pharaoh's heart to be hardened. Rather, it was for the purpose of delivering Israel out by great signs and wonders (Exodus 7:3), and for judgement of Egypt (Exodus 7:4) that a hard-hearted pharaoh ruled.

But Pharaoh shall not hearken unto you, that I may lay my hand upon Egypt, and bring forth mine armies, and my people the children of Israel, out of the land of Egypt by great judgments. (Exodus 7:4)

Therefore, read in this context, "raise thee up" in Exodus 9:16 means "put you in power".

And in very deed for this cause have I raised thee up, for to shew in thee my power; and that my name may be declared throughout all the earth.

How then does this relate to Romans 9? Due to God's foreknowledge, the pharaoh was a "vessel of wrath fitted to destruction" (Rom 9:22) that God suffered long to make His power known. He was not actually a "vessel of mercy" that God had prepared for glory. (Rom 9:23)

What if God, willing to shew his wrath, and to make his power known, endured with much longsuffering the vessels of wrath fitted to destruction: 23 And that he might make known the riches of his glory on the vessels of mercy, which he had afore prepared unto glory, (Rom 9:22-23)

Therefore God was not showing mercy to the pharaoh, but rather the pharaoh (Rom 9:17) was given as an example of "those who he will he hardeneth" (Rom 9:18b). Both the Israelites and the pharaoh were sinners, but one He showed mercy to and the other He allowed to be hardened.

Therefore hath he mercy on whom he will have mercy, and whom he will he hardeneth. (Rom 9:18)

Read Romans 9 in the light of Romans 2:4-11. Mercy is intended to lead to repentance, but in the case of Pharaoh, mercy hardened his heart.

Exodus 8:15

When Pharaoh saw there was relief, however, he hardened his heart...

When God relented and showed mercy, Pharaoh hardened his own heart.

It means God raised up the pharaoh of Egypt up to great power, so that He might show His superior power by defeating the pharaoh; and in order to show His superior power God made sure the pharaoh would resist Him; and Paul says God did this by hardening pharaoh's heart against Him and making pharaoh refuse to agree to His requests, through Moses, to "let my people go".

Not all the pharaohs of Egypt were equally powerful. The fortunes of Egypt ebbed and flowed up and down throughout its history. The Pharaoh God says He has raised up was indeed one of the most powerful in the history of ancient Egypt, or at least he was at the start of his reign, having inherited a crown of immense power from his father, Thutmose III.

According to Scripture Moses was born 1526 BC. Using the High Chronology for the 18th dynasty he was born near the beginning of the reign of Thutmose I. He grew up under Thutmose II, and then Thutmose III became pharaoh who tried to kill Moses after Moses had sided with the Israelite slaves and killed an Egyptian. Nearly 40 years later Thutmose III died and Amenhotep II became pharaoh. Amenhotep II is the one to whom the LORD God says:

"And in very deed for this very reason have I raised you up, to show in you my power; and that my name may be declared throughout all the earth" Exodus 9:16.

Amenhotep II is the pharaoh of the Exodus.

The previous pharaoh, Thutmose III, raised Egypt to the height of its power: he is often referred to as the "Napolean of ancient Egypt" because of his great success on the battlefield. His crowning achievement was victory at the Battle of Megiddo by which he effectively annexed the Levant, the east coast of the Mediterranean Sea (Canaan, Lebanon, etc). So Amenhotep II inherited a position of enormous power from Thutmose III. It is in this sense that God had "raised him up".

But Thutmose III had not won the Battle of Megiddo, nor bequeathed this power to his son Amenhotep II contrary to the will of the LORD but actually it was God who had given these pharaohs their power. And thus Daniel praised God saying

Blessed be the name of God for ever and ever, to whom belong wisdom and might. And He changes the times and the seasons; He removes kings and raises up kings; He gives wisdom to the wise and knowledge to those who have understanding (Daniel 2:20,21).

Now God had raised up Amenhotep II to a position of great power so that Amenhotep II would stubbornly refuse to let the Israelites go. In order for God to be able to show His own power over Amenhotep, Amenhotep was going to have to harden his heart. What if Amenhotep II had said to Moses, "You want all the Hebrew slaves to leave Egypt forever? No problem! You can all go just as soon as you are ready!" In such a case the LORD would not have been able to show His superior power in overcoming the Egyptians: He would not have been able to show His power in the plagues, nor in the parting of the Red Sea: none of it would have been needed. God wanted pharaoh to refuse, to harden his heart. In fact God hardened pharoah's heart for His own glory. So Paul uses this as an example of God hardening someone's heart:

Therefore hath [God] mercy on whom He will have mercy, and whom He will He hardeneth. (Romans 9:18).

God could have chosen to deliver the children of Israel when Egypt, and its pharaoh, were relatively weak. But the LORD did not want to do that. Instead the LORD wanted to display his power against Egypt when Egypt was at the height of its power, so that His own far greater power might be appreciated by His own people and feared by their enemies "that his name might be declared throughout all the earth" (Exodus 9:16), and that He might be reverred.

And so it proved to be, the nations heard how the children of Israel escaped from under the noses of the mighty Egyptians; so that even forty years later Rahab said to the spies sent to view Jericho immediately before the attack in which its walls came tumbling down:

I know that the Lord hath given you the land, and that your terror is fallen upon us, and that all the inhabitants of the land faint because of you.

For we have heard how the Lord dried up the water of the Red Sea for you, when ye came out of Egypt; and what ye did unto the two kings of the Amorites, that were on the other side Jordan, Sihon and Og, whom ye utterly destroyed.

And as soon as we had heard these things, our hearts did melt, neither did there remain any more courage in any man, because of you: for the Lord your God, he is God in heaven above, and in earth beneath (Joshua 2:9-11).

[Moses was 80 years old in the year of the Exodus in 1446 BC. To see how this date is arrived at see: How do Christians reconcile archeology with the Bible in the account of the Battle of Jericho? ]