Why is it important for functions to be anonymous in lambda calculus?
The main theorem regarding this issue is due to a British mathematician from
the end of the 16th century, called William Shakespeare. His best known
paper on the subject is entitled "Romeo and Juliet" was published in
1597, though the research work was conducted a few years earlier,
inspired but such precursors as Arthur Brooke and William Painter.

His main result, stated in Act II. Scene II, is the famous theorem:  


  What's in a name? that which we call a rose
  By any other name would smell as sweet;


This theorem can be intuively understood as "names do not contribute
to meaning".

The greater part of the paper is devoted to an example complementing
the theorem and showing that, even though names contribute no meaning,
they are the source of endless problems.

As pointed out by Shakespeare, names can be changed without changing
meaning, an operation that was later called $\alpha$-conversion by Alonzo
Church and his followers. As a
consequence, it is not necessarily simple to determine what is denoted
by a name. This raises a variety of issues such as developing a
concept of environment where the name-meaning association are
specified, and rules to know what is the current environment when you
try to determine the meaning associated with a name. This baffled
computer scientists for a while, giving rise to technical
difficulties such as the infamous Funarg problem. Environments remain
an issue in some popular programming languages, but it is generally
considered physically unsafe to be more specific, almost as lethal as
the example worked out by Shakespeare in his paper.

This issue is also close to the problems raised in formal language theory,
when alphabets and formal systems have to be defined up to an
isomorphism, so as to underscore that the symbols of the alphabets are
abstract entities, independent of how they "materialize" as elements
from some set.

This major result by Shakespeare shows also that science was then diverging
from magic and religion, where a being or a meaning may have a true name.

The conclusion of all this is that for theoretical work, it is often
more convenient not to be encumbered by names, even though it may feel
simpler for practical work and everyday life. But recall that not
everyone called Mom is your mother.

Note:
The issue was addressed more recently by the 20th century American logician
Gertrude Stein. However, her mathematician colleagues are still
pondering the precise technical implications of her main theorem:  


  Rose is a rose is a rose is a rose.  


published in 1913 in a short communication entitled "Sacred Emily".