I honestly think that the "we found oil" answer is the simplest, but in the interests of giving an alternative, I'll answer something different.
IMHO one of the wisest statements is this (paraphrased): "You do not need to run faster than the bear, just faster than the slowest of your friends."
Basically, if your country is the best place to do business in the area, that is what people will do. You do not have to solve your issues, if the other countries you directly compete with for investment have the same issues worse.
Establish strong rule of law to curb issues with dictatorial rule. People do not really care about the level of democracy, what they care about is that dictatorships are arbitrary and unpredictable. That makes investments riskier, which limits investments to the most profitable. It also makes living in the country riskier, which means that valuable people with skills are likely to come to the country or stay in the country only if they have no good alternatives.
This is about reforming the legal system. Reform the courts. Reform the laws. Then make clear that you respect the laws and the legal system. Aim to be remembered as "the Just" and commit to that publicly.
If you succeed, you will also be able to open the borders for people coming in or going out, which makes the economy more competitive.
Make fighting corruption your priority with a laser-like focus. Invest real resources to cleaning up corruption long term. Accept that the corruption in your country is systemic, not something caused by individual criminals. Be lenient with people who confess, cooperate with the investigation, and promise to clean up. Make examples of people who lie to your investigators or the courts, or break their promise of going clean. Protect and reward people who reveal corruption, punish people who knew of corruption, but kept silent (this needs to be introduced gradually).
Reform the religion in your own image. You do not wish to be "the Just" because it will improve the economy. Oh no, you believe that God is the God of Justice, and to be Just is to serve God and to be Unjust is to fail God.
Fight against corruption is a Holy War against the forces of darkness. Corruption is not about money, it is about the souls of the people turning away from Justice, the path to God, towards Injustice, the path towards eternal damnation.
And the forces of darkness have not been content to corrupt the secular, they have even spread their insidious poison to the very religion that is meant to protect the souls of the people from them and guide people to God. Religion needs to be cleansed, refocussed on the Justice of God, against the Corruption of the Darkness.
No more will people be distracted from true path to God with calls to the dark sides of the human souls. No longer will the word of God be allowed to be used to spread hatred and intolerance. No longer can word of God be used to spread fear of Change when the God is omnipotent and no change cannot happen that is not within his power. To fear change is to reject the very power of God.
You get drift? Basically, even though "the Just" is the epithet you are going for, you should play so that you will probably actually be remembered as "the Holy" or "the Pious". Such brute force solutions are really the best for dealing with religious issues after a certain point. The religious actually want to be told how they should believe. Give it to them. Validate their view that God and religion matter and they will love you.
Just remember that dogma and absolutism only makes sense on the level of principles, when it comes to actual solutions humility and conversation are what you should preach. Corruption is Evil and from the Dark and must be destroyed without mercy. Corrupt people are victims of the forces of Darkness, and should be guided back to light with understanding and negotiation. Unless they reject the guidance in which case the iron fist comes to play.
Brotherhood of faith is how you see your neighbours. The economy of Britain and Ireland has done well not just because they have been good places to do business, but because they have been good places to do business within the EU. More of their business within the region business can do within your country more money you will make. Because of this you should do everything you can to promote regional unity and cooperation. Be the good guy, take the high road if you can.
That said in practice you should be pretty Machiavellian. Remember that the world is not actually just and act accordingly. Even if you are fighting corruption, your neighbours will still be riddled with it. Accept it and take advantage of it. High corruption means your neighbours will have powerful men with lots of money they can't really invest in their own country. (Because... you know... with the high corruption...) Mutually beneficial arrangements where they invest in your safe country to get benefits in their own country should be pursued.
If your poor country does not produce something, but their country does, it is trivial to open the market on that good and then move the production to your country with its lower corruption. You can even agree to free movement for the employees in such cases, so the owner of the factory can even keep his core trained work force. This is simple profit for the owner of the production and since in corrupt systems he would also have influence on the people who make the decisions on the issue it should be simple to arrange. In some countries people owning the business and making the decisions might even be the same.
I was reading the comments to the "natural resources" answer and realized I forgot to say some things. First, and rather obvious, this answer is not only an alternative to natural resources, it works even better as a supplemental solution. "What to do when you find oil." Second, this and the "natural resources" answer tell how to get the growth started, but I forgot to tell one tip about sustaining the growth.
Think small and iterate. Instead of banking on one industry or product, you should work to create as many healthy self-sustaining industries and services as you can. Do not aim to be number one in one thing, aim to be solid in many things.
Economy forms a network and can form self-sustaining loops. Because of this it is very valuable to have a diverse economy, so that those connections and feedbacks can happen in your country. Otherwise part of the growth will happen elsewhere.
Businesses might even relocate to a location that has all the things they want, because while transport has become cheap it still adds lag. New growth and investment tends to happen in places that have most of the necessary resources and services available because quite simply it makes starting new business easier and faster.
Importantly pushing this threshold low enough is the requirement for organic growth. (<- What you want.) Easy availability of resources and services lowers the threshold. Corrupt and inefficient administration raises it.
The importance of availability is actually part of the argument why free trade is supposed to create economic opportunities and growth. This argument definitely applies to a poor country trying to become wealthy. Even if a natural resource or foreign investment gives you the capital, you won't have the volume or diversity to sustain healthy economic growth. You will need a conscious policy of sharing these resources with others in the region and even globally. Your country should be open to foreign companies and your companies should be aiming to operate internationally.
Two good starting points to think about are services and infrastructure. My answer about reducing corruption and improving administration is a good starting point for becoming the regional service hub. The primary answer about natural resources is a good starting point for making capital heavy infrastructure investments. Typically we would be talking about transport infrastructure, but Las Vegas (among others) is an example of spending money to create a new travel destination. Money has also been spent on new capitals, huge vanity projects, and investment funds, but the relevance of those to this question is not obvious.