How can I rehabilitate the grout in my tile floor?
If the grouting wasn't well done (or was OK initially, but then not sealed), I would be tempted to remove it with a grout saw and redo it properly.

Whether you should do just the part affected or the whole lot is a question of whether you want to repeat the exercise for the part you didn't do in the future. These types of jobs half done, or in this case half re-done, always tend to come back to haunt you. As an example, if the part nearest the doorway is worst affected, then likely it's a problem of moisture/water from shoes on entry. And you therefore might be able to get away with just redoing that portion.

As for using a colorant, this might work, but if the grout is in general moisture affected, i.e., now not properly bonded, it's could likely come out at some point in the future anyway...

Hope this helps!