Every GP is very precious and you need a lot of consideration to figure out how to use him the best way possible.
Build the Shrine - if you have a Holy City you can use Great Prophet to build a centre of religious cult. Shrine gives culture bonus and helps to spread religion, but first of all gives a lot of money: +1 Gold for every city where particular religion's believers are present. What is very important - this is working not only for your state religion, so we can earn money on more than one religion.
Super specialist priest create: +2 Production, +5 Gold
Hurry - this function gives immediately approximately 1000 Production to any building. This makes it possible to finish most of buildings and Wonders in one turn. Very important - Great Engineer can only hurry buildings and wonders, not projects, so most of very advanced structures like spaceships parts or Internet, SDI, Manhattan Project cannot be hurried.
Super specialist engineer create: +3 Production, +3 Research
Trade mission - this is a very significant money bonus. You send a Great Merchant into a big (the bigger - the better) and distant (the more distant - the better) foreign city. Amount of cash made by trade mission varies on these two factors - that's why it's a good idea to look for the best city to use this Great Person and check a lot of potential places where GM can make his deal. Because income form trade mission is dependent on city size amount of Gold become higher as game proceed. In early-game is usually 700-800 GP, in mid-game good deal is 1500, later it's possible to receive 3500 or even more.
Great Merchant can also work as a great spy - he can come into foreign territory even if we don't have open borders with this civilization.
Super specialist merchant create: +1 Food, +6 Gold
Build Academy - this building gives the city +50% research bonus. It is very significant - especially in early-game, and if you build Academy in suitable place it is a perfect long-term investment. A suitable place means that city can generate large amounts of research now (due to specialists or lot of commerce) or will be developed as a research centre in the future.
Super specialist scientist create: + 1 Production , +6 research
Great Work or the culture bomb - It instantly makes 4000 culture points, what usually causes a major border change, what can effect in securing access to important resources or foreign border city flips. If you call the culture a peaceful way of expansion, than culture bomb is a peaceful blitzkrieg.
The second gain from the culture bomb is immediately stopping any resistance or riots in the city ad gives city +60% defence bonus - what makes culture bombing extremely useful in time of not-so-peaceful expansion. Making already conquered enemy city's inhabitants your extremely loyal citizens helps a lot when you need a secure base of operation on enemy territory.
Super specialist artist create: +3 Gold, +12 Culture
Every Great Person can be also used in two other ways.
First one is researching technology - major boost of researching one advancement correlated with speciality of our GP. In case of early-game technologies means immediate research of this technology. In situation of medium and late game technologies it is merely a major bonus to research - sometimes useful, sometimes not.
One tip - if you don't like technology that GP can research, because it is very easy to research it in standard way or easy it's to buy from another civilization you simply should do this and after that GP receives a new advance to research.
The second option is triggering the Golden Age - the eight turns time when your Production and Commerce output is significantly enhanced. However usefulness of the Golden Age is limited. It is only helpful in situations when we really need an edge over opponents (like in time of total war or space racing) because usually bonuses are not so significant to justify their costs. Every time you trigger the Golden Age it costs more GP. First it will be only two GP of different kinds, next time three and so on.
As a side note, it is worth to mention that we can also trigger the GA by building a Taj Mahal Wonder - which is much less expensive and far more useful way of introducing Golden Age.