I'm wondering if Earth would try to keep any colonies we put on other celestial bodies in this solar system to be economically dependent on the mother planet.
Earth won't feel any need to try
Given our current level of technology, it is quite plausible that we could send a manned mission to any of the planets in our solar system. The problem is that it would be a one-way mission — it would take significantly larger rockets with a lot more fuel in them (or multiple trips putting fuel into orbit) in order to get to space with enough left over to get to another planet and back. Still, if there was another habitable planet in our solar system, it would be reasonably easy for us to get people there.
However, as you should be aware, there aren't any other habitable planets in our solar system. Both Mercury and Venus are too hot, and everything from Mars on out is too cold. Any colonies on the other planets in our solar system will require high-tech solutions in order to simply survive.
Because of this, any extraterrestrial colonies will be heavily dependent on the Earth for quite a while. What will it take for them to not be dependent? They'll need to be capable of manufacturing everything they need for their survival and for growth.
Let's use Mars as a concrete example, as it is probably the least hostile. It's atmosphere is not breathable — way too thin, and way too much CO2 in it — but it could be compressed and added to the air in a greenhouse in moderate amounts. This would increase the total air supply available to the colony. The soil is reasonably close to what plants need to grow, and it wouldn't be too hard to either bio-engineer plants that can handle Martian soil and/or treat the soil.
The biggest obstacles will be getting enough electricity in order to process enough raw materials to be able to enlarge existing structures and build new ones. Iron is quite common, but smelting pure iron from ore takes quite a bit of energy. At first this will happen using solar energy, but when they can get nuclear energy going things will go much faster (now that I think of it, it would make sense for them to design a combined nuclear reactor and smelter). There's uranium on Mars, so it doesn't have to be imported from Earth, but without being able to just pump regular atmosphere down into mines, mining will be harder than it is on Earth.
All hydrocarbons (oil, plastic, rubber) will need to be made the hard way — taking organic material and processing it.
It's likely that colonies will expand primarily underground where you can build without needing to use significant amounts of airtight materials.
Now that I've shown some of the reasons why it will be hard for an extraterrestrial colony to be self-sufficient, how can the Earth "profit" from owning such a colony?
It's going to take a significant amount of time and/or imported materials in order to get enough infrastructure set up for the colonies to be self sufficient, making for huge upfront costs. Even then, what economic or political advantages could a colony provide? Anything that could be produced on Mars could be produced on Earth, and even if Mars' lower gravity made it easier to produce the shipping cost would easily overwhelm that advantage.
Also, suppose that the colonists on Mars figure out how to make something new that we don't have on Earth yet? As I've said, it will be quite expensive for them to expand their available area. Even with a head start of a decade, it would not take Earth long at all to reverse engineer whatever it is, figure out how to manufacture it, and build up even larger factories to produce it at a much faster rate.
One thing that could tip the balance in Mars' favor is if the planet has been terraformed. However, that will be an absolutely massive undertaking - with its atmosphere currently at 1% the density of Earth's, an enormous amount of gas will need to be produced. That undertaking will take millennia - easily long enough for any ownership issues to be worked out.
A colony doesn't provide any political power, either — suppose China had a colony on Mars right now, and the US and China got into a war (whether military, trade, or whatever). What's the colony going to do? Shake their fists at us?
There are useful things that Mars can provide, but none of them are really dependent on a colony belonging to Earth. If humanity continues to grow, we'll eventually start running out of room on Earth. Living space on Earth could become so expensive that moving to Mars is actually reasonable. This would work best if there is already a colony on Mars, but it doesn't really matter if that colony belongs to a particular Earth country or corporation or not.
Another possibility is inventions. I've already mentioned some of the difficulties I can foresee a Martian colony needing to overcome. In particular with hydrocarbons, there will be quite a bit of interest in developing new strategies to process waste biomatter into useful hydrocarbons. These new techniques could prove quite useful on Earth as well.
However, none of these things require being on Mars, just going there. The US space program has resulted in quite a few useful inventions (such as velcro), but they have been a result of solving the problems of going to space rather than some product of being there. Similarly, I believe that going to Mars or other planets will result in a number of very useful inventions, but none of them will actually require anything from the other planets.