Could a species of aquatic mammals develop bio-luminescence? What kind of bio-luminescence would it be? What circumstances could lead to it?
Yes, why wouldn't it be?
Just look at other aquatic animals which have developed bioluminescence
Just because a mammal is a mammal doesn't mean it couldn't itself evolve bioluminescence.
To see how mammals could accomplish bioluminescence, look at how other species do it: the anglerfish isn't bioluminescent on its own, instead it lives in a symbiotic relationship with some bioluminescent bacteria.
Now why couldn't mammals exhibit bioluminescence via a similar method?
If the aquatic mammal were to have luminescent bacteria, algae or perhaps some phytoplankton growing from it then it would be bioluminescent.
The glowing organism doesn't even need to cooperate! Your mammal could literally just swim through some phytoplankton and become bioluminescent, albeit temporarily, until the plankton dies.
If the mammal wanted to exhibit some permanent bioluminescence, via symbiotic means, then it would have to benefit the parasite in some way: perhaps it could have sugar glands in patterns along its skin, attracting a certain species of bioluminescent bacteria.