Question: Assume there was a super advanced alien race that for some reason, wanted to add a new planet, of lets say 1 Earth mass, into the Solar System. Where in the inner Solar System *(The part of the solar system inside Jupiter's orbit, so asteroid belt is included)* could we put this planet, so that it would have a stable orbit for billions of years?

*Note: I would preferably not want have this planet orbit closer to the sun than Mercury, and would like for it to affect the inner planets as little as possible, although I care more about the first condition rather than the second.*

*And also, the planet needn't be habitable*

Answer: 
If you have the resources to create or bring a new planet - generally, wherever you feel like. You will be probably able to maintain the orbital stability as well.

On the other hand, if you are limited in long-term expenses, the orbital stability becomes an issue. And there, the answer becomes "nowhere in the habitable zone".

Mercury, Venus, Earth and Mars are in a complex orbital relationship, maintained by constant exchange of momentum between the planets. There are hints that the solar cycle is related to the tides the first 3 planets exert at the Sun as well.

How can one save the plot, then?

* Replace Venus / terraform Venus. It is Earth-mass and with some added water and without the greenhouse effect can be pretty much habitable. And only slightly hotter than the Earth (if at all).
* Make Mars double. Mars is not much different than the Moon in both mass and composition. Put an Earth-sized planet there and allow Mars to orbit around. A heavier body in the place of Mars will likely have only minor effect on the Earth's orbit. The two Mars moons can be put in an orbital resonance with Mars (3-moon world?), mined into oblivion or stored in the asteroid belt until better ideas arise.
* A combination thereof (replace Venus, store Venus near Mars where it can cool off for a while).

Beyond Mars it's rather cold, but if you don't mind that, the asteroid belt itself is a place. Not a place to have a planet on its own, but pretty much a place where a planet would be stable. Vacuum it first or prepare for a serious meteorite shower later. You can use the collected material for a little fancy moon as well.

Beyond the asteroid belt it is really, really cold, in particular too cold for almost any type of atmosphere or hydrosphere. If you can bear the heating bill, neither Jupiter, nor Saturn, nor the ice giants beyond, will mind an Earth mass satellite. You will have to rearrange the moon zoo there first in order to avoid crashes.

Don't put anything heavy below Venus orbit even if you don't mind the heat. We like our calendar and seasons pretty much. Thank you in advance.
The planet would be able to be the most stable in a orbit between earth and Mars because that the biggest gap between notable celestial bodies of the inner solar system. So gravitation pull of any bodies would be most minimal hence being able to keep a steady orbit as long as the sun lives.

Edit:I apologize by I messed up(dyslexic). It rather crowded between mars and the belt you best bet would be between earth and mars.

Another solution would to have an orbit perpendicular of all the other orbit since most bodies in this system tend to stay on one plane.