Stephen is the son of Lady Britomart and Andrew Undershaft. He tries pretty hard to be an ally for his mother, initially refusing to be anything more than civil to his father when he comes around and charms Barbara and Sarah (and their fiancés, of course). Stephen lets his mother bully him into "giving" her the advice to ask Andrew for money, even though he is personally against accepting any further money from his father. He puts up a bit of a fight, but ultimately he just gives in to his momma. He starts out the play pretty much entirely under Lady B's thumb.
Poor Stephen doesn't get much more respect from his father at times, either. For example, when his father is trying to figure out how to help Stephen into a career that doesn't involve the Undershaft business, he gets annoyed at Stephen's pretentions to honor and morality. When Lady Brit asks what he thinks Stephen should do for a living, Andrew sneers back:
Oh, just what he wants to do. He knows nothing and thinks he knows everything. That points clearly to a political career. Get him a private secretaryship to someone who can get him an Under Secretaryship; and then leave him alone. He will find his natural and proper place in the end on the Treasury Bench. (3.127)
Luckily, by the end of the play, Stephen ends up in a better place with both parents…from his perspective, at least. First off, he finally stands up to his mom and tells her to butt out of his decision not to press to take over the family business:
Mother: there must be an end of treating me as a child, if you please...Until last night I did not take your attitude seriously, because I did not think you meant it seriously. But I find now that you left me in the dark as to matters which you should have explained to me years ago. I am extremely hurt and offended. Any further discussion of my intentions had better take place with my father, as between one man and another. (3.107)
Second, even though he had been steadfastly against his father's business throughout the entire play, he ends up thinking the Undershaft factory is pretty great when he visits it. When his father asks his opinion, he exclaims:
Oh, magnificent. A perfect triumph of modern industry. Frankly, my dear father, I have been a fool: I had no idea of what it all meant: of the wonderful forethought, the power of organization, the administrative capacity, the financial genius, the colossal capital it represents. (3.202)
So, basically, like everyone else, Stephen ends up seeing what a swell guy his dad is, and mans up in the process. Hooray?