This seemingly straightforward question is a little complicated because the answer all depends on whom you believe offered the challeng. In Act 2 of Pygmalion, Liza shows up at Higgins' doorstep much to his outraged surprise ("shall we throw her out of the window?") and Mrs. Pearce's well-mannered distress ("Very common indeed. I should have sent her away"). Liza comes to Professor Higgins to offer a challenge to him. The challenge is to take her on as a student and teach her proper English so that she can attain a position in a flower shop instead of pursuing the necessity of selling flowers in Tottenham Court Road. Higgins calculates the price she's offering for the lessons on a percentage of daily wealth, contemplates the greatness of the end result and then takes her up on her challenge: "Yes: in six months ... I'll take her anywhere and pass her off as anything."
Liza, though this is what she wants, balks a little at some of his conditions, like sleeping among black beetles if she doesn't learn her lessons well and having her clothes burned and being beheaded and such. Now comes Higgins turn to try to persuade her. I wouldn't exactly say that he offers Liza a challenge; it is much more like a bribe (anything given with the view to persuade or induce, to corrupt actions away from moral or normal choice), though some may choose to refer to it as a challenge (the call to accept a stimulating undertaking or a call to a contest of skill). After Liza has hollered "Ah-ah-ah-ow-ow-oo-oo!" a good number of times and rightly protested that he is "no gentleman, youre not," Higgins' eyes alight on the perfect "challenge" resting atop the piano and offers Liza a wage for her work on language with which to buy herself all the chocolates she wants. Thus inspired, he remembers her delight and pride in having arrived in a taxi and adds daily taxi rides to the "challenge"--or bribe--as you will.
This "challenge" strikes Liza favorably and she gives over her protests against being washed up ("Ah-ah-ah-ah-ow-ow-oo-oo!!! I aint dirty: I washed my face and hands afore I come, I did"), having her clothes burned and sleeping amongst black beetles. The challenge is on: Higgins will teach Liza, and Liza will suffer to be taught by Higgins. Finito--or the beginning.