Yes, the gaze attack affects invisible creatures.
The gaze attack is automatic as long as you are looking at the creature with the gaze attack. In this case, in order to avoid the bodak's death gaze, the invisible attacker would need to avert his eyes on his turn. It doesn't matter if he's invisible or not, because the ability triggers when you look at the creature who has the gaze attack, not when the creature with the gaze attack looks at you.
Each character within range of a gaze attack must attempt a saving throw (which can be a Fortitude or Will save) each round at the beginning of his turn.
An opponent can avert his eyes from the creature’s face, looking at the creature’s body, watching its shadow, or tracking the creature in a reflective surface. Each round, the opponent has a 50% chance of not having to make a saving throw. The creature with the gaze attack gains concealment relative to the opponent. An opponent can shut his eyes, turn his back on the creature, or wear a blindfold. In these cases, the opponent does not need to make a saving throw. The creature with the gaze attack gains total concealment relative to the opponent. (Source: d20 SRD: Special Abilities)
At the end of the section on gaze attacks, it also says this:
Unless specified otherwise, a creature with a gaze attack can control its gaze attack and “turn it off ” when so desired. Allies of a creature with a gaze attack might be affected. All the creature’s allies are considered to be averting their eyes from the creature with the gaze attack, and have a 50% chance to not need to make a saving throw against the gaze attack each round.
This means the bodak chooses when it is on.
Long story short, if you can see the bodak, you have the potential to be affected, regardless of your invisibility.