1. Hillary is the one who rescued the bird.
2. Bill is on the team that won first place.
3. She belongs to an organization that specializes in saving endangered species.
However, the improper use of "that" for "who" when referring to a person, seems to be increasing all the time. There are more than 700 Google News hits for the phrase "the person that," including the following examples, mostly of quotes within a news article:
"The person that made the call..."
"I loved her for the person that she was...."
"The person that donated the money..."
"The person that is causing the problems...."
In all those cases, I think it should be "The person who... "
Likewise, there are more than 1,000 Google News hits for the phrase "the man that," such as "...the man that the left hates the most, President Bush..," which I suggest should be "The man who..."
Finally, here's an example of using both "that" and "who" in the same sentence! "The quarterback that lost fumbled and threw three interceptions. The quarterback who won, though, is the one who got pulled on Saturday."
Update: When you use "grammar check" in Microsoft Word for the sentence "She is the one that found the bird," it accepts the "incorrect" sentence as written, and of course it also accepts the correct sentence "She is the one who found the bird."