Can any computably enumerable set be generated by a prefix-free set?
After a lot of thinking and reading, and getting a helpful answer and comments from Andrej Bauer to another question that my investigation prompted, I can answer my own question.  I post an answer for the benefit of anyone who may find the question to be similar to one of their own.  I expect no upvotes for answering my own question.
Downey and Hirschfeldt prove (2.19.2, p. 74) that every $\Sigma^0_1$ set of infinite sequences is one that can be generated by a c.e. set of finite strings.  Moreover, they define Martin-Löf randomness in terms of a sequence of $\Sigma^0_1$ sets $U_n$ of infinite sequences.  This is why they have the right to assume that every such $U_n$ can be generated by such a set of finite strings.
In my gloss of D&amp;H's description of a Martin-Löf test, I stated the requirement that the test sets be $\Sigma^0_1$ as a requirement that they be computably enumerable.  One can see the equivalence of $\Sigma^0_1$ and c.e. as implied by D&amp;H's proposition 2.19.2, but it's proved directly by, for example, Nies, 1.4.12, p. 22.  So the way that I characterized D&amp;H's description of Martin-Löf tests was correct.
While it's true that $U_n=\{000\ldots\}$ can't be generated by finite strings, my mistake was thinking that such a $U_n$ is computably enumerable.  It was surprising to me to realize that such a trivially simple set is not c.e.  After all, the set has only one element, and a Turing machine that generates it or checks for it is trivial.  The crucial point, though, is that that machine cannot halt on $000\ldots$, since the sequence of zeros is infinite.  No program can ever successfully list or accept even (the) one member of this set.  Thus my sequence of sets $U_n$ do not form a Martin-Löf test.
(It is possible to define a Martin-Löf test that excludes only $000\ldots$ from the random sequences, but that test has to consist of sets such as, for example, $U_n=\{x:$ the first $n$ digits of $x$ are 0$\}$.  Each such set contains an uncountably infinite number of infinite sequences, but each is a subset of previous sets $U_1, U_2, \ldots, U_{n-1}$. The one sequence contained in each of them is $000\ldots$ .)