Why is that used in the following sentence, instead of who?
Why was the girl that had plenty of money arrested for shoplifting some trinkets worth only about two dollars?
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This response is actually directed to the comments that appear beneath waywardEevee's answer—and in particular to the OP's remark that "Why was the girl that had plenty of money arrested for shoplifting some trinkets worth only about two dollars?" was marked as the correct answer on a certification test for English teachers.
As John Lawler points out in his comments, claiming that either who or that is THE correct answer is arbitrary and ultimately baseless, since either word can validly be used to introduce the dependent clause "had plenty of money."
But whereas Araucaria sees the test designer's preference for that as perhaps striking a blow against an enforced blanket preference for who in all instances involving human referents (such as "girl" in the example sentence), I see it as a survival of another (and even older) arbitrary rule of usage that may be even less firmly connected to real-world usage. This rule asserts that that should be used to introduce all dependent clauses of the that/which/who type, whether the reference is to a person or to a thing; that who should be used only to introduce independent clauses that refer to a person; and that which should be used only to introduce independent clauses that refer to a thing.
Here, according to Frank Vizetelly, A Desk-Book of Errors in English: Including Notes on Colloquialisms and Slang to Be Avoided in Conversation (1908), is how the rule works in practice:
that, who : Discriminate carefully between these words. That implies restriction; who generally denotes coordination. As an illustration of this distinction, Alfred Ayers says ("The Verbalist ," p. 202), "'I met the boatman who took me across the ferry.' If who is the proper word here, the meaning is 'I met the boatman, and he took me across the ferry,' it being supposed that the boatman is known and definite. But if there be several boatmen, and I wish to indicate one in particular, by the circumstance that he had taken me across the ferry, I should use that." That ought, therefore, to be preferred to who or which whenever an antecedent not otherwise limited is to be restricted by the relative clause.
It seems to me that the test designers marked that as the correct answer in the sentence about the affluent shoplifter because they were enforcing the 1881 Alfred Ayers rule for dealing with that/which/who constructions—an exercise in grammatical atavism that no one ought to applaud.
Who vs that is kind of a grey area. The general rule is who should be used when referring to a person, that where referring to an object. In practice, though, that has been used as a relative pronoun when referring to an individual for a long, long time.
I think it's common to use who when the clause is being used to identify the person, and that when it's used descriptively. E.g.
The girl who had plenty of money was arrested.
implies that there were several girls who might have been arrested, but only the one with plenty of money actually was. You would use the above sentence to answer the quesetion Which girl was arrested?
The girl that had plenty of money was arrested.
says nothing about whether other girls were involved. It just means that a girl was arrested, and she also had plenty of money.
However, these are subtle nuances, and not all speakers would obey them. Often the context will make it clear which was intended.