Jim Kelliher
The only exercises I feel competent to recommend are those I have done myself, which means I have only a few sources of exercises. Two other obvious sources are Ahlfors and Palka, being on the syllabus, but I have worked only a few exercises from those books.
Functions of One Complex Variable I, Second Edition by John B. Conway
As a general rule Conway's exercises are easy for a graduate text, but many of his easiest exercises contain not-so-obvious facts that are useful for solving many other, harder problems. The exercises within a section tend to get progressively harder and to build on previous exercises, which is often not the case with Rudin. At least one of Conway's exercises appeared verbatim on a prelim, and some of them are similar to past prelim problems.
Real and Complex Analysis, Third Edition by the master, Walter Rudin
This is the most beautiful math textbook intended for the classroom ever written. (I ignore the sharp cries of disagreement.) Prelim problems are often taken directly from it, and many of its exercises are considerably harder than any prelim problem not appearing on the August 2001 prelim. Few people can safely ignore Rudin and hope to pass UT's analysis prelim. It is particularly important to work all or at least nearly all of the problems in the first three chapters and in chapter 10. This alone is a project for a normal summer--which a prelim summer is not--but one cannot ignore chapters 6, 7, 8, and 12, so some compromise has to be made. I have taken a shot at selecting problems from Rudin, reluctant as I am to in any way be seen as editing the master.
Problems and Theorems in Analysis I by George Polya and Gabor Szego (umlauts and accents suppressed)
Problems 266-279 on the maximum modulus theorem are worth working (as, I am sure, are many of the other problems, but I haven't worked them). The notation and terminology in this text are a little different (older?) than those of Conway or Rudin, and the wording is often clumsy, possibly due to the translation, making some of the problems hard to understand.
Measure and Integral by Wheeden and Zygmund
I don't have any recommendations for exercises from this book. I mention it because I prefer their discussions of convolution and of functions of bounded variation and of absolutely continuous functions to that of Rudin's. Also, their presentation of differentiation and of integration on product spaces complements Rudin's well. Their discussion of Fubini's theorem and of Tonelli's theorem (which Rudin never names) is more hard-nosed and clearer.
Other exercises
Two exercises on absolute continuity and bounded variation, extracted from Dr. Vaaler's Spring 2002 Fourier Analysis class.