Can 'human' become a genus due to space colonization?

I do not believe it will happen. There are multiple roadblocks:

First, speciation time is measured in generations, not years. The human generation time is long, the 3000 generations mentioned in another answer for a fish translates to nearly 100,000 years. Are human populations not going to interbreed over a time span that long?

Second, the speed of speciation is a function of the evolutionary pressure. We radically alter our environment and change the whole driving force--genes which make one more or less able to survive the rigors of the environment have almost no effect anymore. To the extent that human evolution will occur it will be mostly towards things that favor reproductive success--things which attract mates, a desire for children, contraceptive unreliability. These things aren't going to be substantially different on Mars than on Earth.

Finally, we are already to the point that we can make designer babies to some degree, although we can't yet do it with the sort of reliability needed to ethically do it on humans. I find it inconceivable that we can remain a technological species (as would be required to have a population on Mars) without reaching the point that we can freely rewrite the genetic code at least during reproduction. If there is a speciation event it will come from the lab, not evolution. It's possible that some day there will be a modification developed that is a substantial upgrade but which is not compatible with existing humans--those with the modification can only breed with others that have it.

As others have noted, the key concept is speciation. Imagine some H. sapiens living on Mars. How are they going to breathe? Presumably we have a nuclear reactor generating power, which in turn is used to make oxygen. What is it breaks down? Those who can tolerate low oxygen levels might do better. Here on Earth we invest a lot of energy in building bones and things to resist gravity. On Mars there may be less need to do this, and more need to be efficient with oxygen and tolerate higher levels of CO2. So some mutations that would be sub-optimal on Earth might be beneficial to the colonists. Couple this with isolation, as others have mentioned, and the two populations may drift apart, leading to the eventual emergence of H martiansis (or whatever.).

The concept you are referring to is speciation and it has been well studied in a wide variety of different natural organisms. I suppose here we are talking about the biological species concept.

The overall answer is yes it is possible, but critically depends on a few different factors. The reality of speciation in the wild is very complex, but these are some things to consider:

Genetic isolation

If two groups, such as your Martian colony and humans on earth isolate from one another, then for speciation to occur, there needs to be a significant level of genetic differentiation between them. This means substantial differences in the kinds of genetic variants found at positions along the genome.

Genetic differences are eroded by post-isolation gene flow – in your example, that might mean a spaceship flying back to Earth with Martian colonists, who then have offspring with people on Earth.

Although there is plenty of evidence that speciation with gene flow can occur, the general rule of thumb is that increased gene flow means a longer divergence time is required to fully speciate.

In nature, this is often caused by some kind of geographic barrier to gene flow, such as mountains or rivers forming, but it can also be caused by morphological differences, such as variation in sexual appendages. Of course, in our example, this barrier to gene-flow would be the large and difficult to traverse distance between the Earth and Mars.

Divergence time

You alluded to some kind of separation time, and you are right to be talking on the scale of millions of years. Speciation can occur extremely quickly; in Lake Tanganyika cichlids, it has occurred probably within the last 15,000 years. Humans have created whole new species of crops, such as maize, within the past 10,000 years. There is even evidence of some fish speciating in 3000 generations.

However speciation is often a much longer process. For example humans and chimps were thought to speciate in 4.5 millions years.

Of course, there is an interaction with several other factors. All other things being equal, less post-isolation gene flows results in a shorter time to speciation and vice versa. Stronger selective pressure between the different environments leads to a more rapid accumulation of genetic differences.

As Konrad Rudolph correctly points out, divergence time is strongly related to generation time, with all other things being equal, a shorter generation time, results in faster speciation.

Selection pressures & differing environments

I think the last main factor governing speciation is how different the Martian colonists environment was from that on Earth.

Different environments can lead to natural selection occurring in opposing directions in the two populations, leading to ecological speciation. Speciation can proceed without differing environments, where neutral drift in allele frequencies can eventually cause speciation, but this will be a long process.

Speciation will occur much more rapidly if there is a start difference in environment and selection pressures between the two groups.

So in conclusion, given enough time, genetic isolation and differential selection pressures (or some combination of the above), it is plausible that a new species of human could form. However, given the time-scales required, it seems a bit unlikely to me.

EDIT

I think it is worth pointing out what @Jaquez said in the comments. If current terrestrial humans split from an extraterrestrial source to form another species, it would be named as another species within the Homo, such as Homo extraterrestrialis, for example. The addition of a new species does not move the group Homo up to become an e.g. Tribe or Family.