I know that you can combine incompletel proteins in plant based food
to create a complete protein - eg., legumes and rice, nuts and grains.
 Also, I know you can combine an incomplete plant protein with a small
amount of animal protein, and make a complete protein from the plant
source.

However!  I cannot find any information on whether an incomplete plant
protein is of any use to your body by itself, e.g., not combined with
another protein source.

For example, if I eat a bowl of black beans for breakfast, and no
other protein sources at all the rest of the day (and this is an
extreme and perhaps unpalatable example, I know), will it do my body
any good, especially with regard to the protein?

Interesting question. Foods with incomplete proteins are not
nutritionally worthless, but their "biological value" and ability to
support growth are reduced when compared to foods (or combinations of
foods) which supply a full complement of amino acids. Incomplete
proteins are not capable of building muscle, as complete proteins can;
a food with incomplete proteins, unless eaten with a complementary
food which supplies the missing amino acid(s), acts more like a fatty
food or a carbohydrate-laden food, since it will primarily be used for
fuel, not for cellular repair.

In your example (eating a bowl of black beans for breakfast, and no
other protein for the rest of the day), the incomplete protein from
the beans would not function as a tissue-builder for the body, so the
protein would, in a sense, be "wasted." However, the beans would
provide energy and fiber, so it wouldn't be as if you had eaten a bowl
of Kleenex for breakfast.

"To be useful to a person, the totality of food proteins must be
'complete' - that is, all eight essential amino acids must be ingested
simultaneously, and in the right proportion. Incomplete proteins
cannot be used to build muscle and tissue; they often end up as stored
fat or are utilized for energy...

'Certainly some vegetable proteins, if fed as the sole source of
protein, are of relatively low value for promoting growth,' the
editors of the British medical journal Lancet wrote in 1959."

A STUDY GUIDE TO COUNSELS ON DIET AND FOODS
http://www.whiteestate.org/guides/CD.pdf

"Protein requirements are also based on the protein quality, as
measured by the biological value (BV). Protein is also measured by the
way it supports growth; this measurement, called the protein
efficiency ratio (PER), is determined by feeding an animal a
particular protein food and measuring its growth.

The reference protein for determining the biological value of foods is
that of eggs (ovalbumin), the food with the highest BV at 94 percent
(although mother's milk is valued at 100 percent). Next are fish at
75-90 percent, rice at 86 percent, legumes at 70-80 percent, and meats
and poultry at 75-85 percent. Corn, an incomplete protein, has
approximately 40 percent biological value."

HealthWorld Online: Amino Acids
http://www.healthy.net/scr/Article.asp?Id=2108&action=print

"Protein foods are classified in two ways: complete and incomplete.
Complete proteins, which come from animal sources such as chicken,
fish, dairy and soybeans, contain all the essential amino acids that
help build your muscle and body tissue. Incomplete proteins, found in
plant foods, such as grains, seeds, nuts, beans and vegetables,
provide a varying but limited array of amino acids. A greater variety
and amount of incomplete proteins must be consumed to cover all the
amino acids needed for protein building.

We can compensate for the amino acid deficiencies in an incomplete
protein by combining it with another protein, thus providing all the
building blocks for protein creation. This is the concept of
complementary proteins, in which proteins with opposite strengths and
weaknesses complement each other.

For example, many cereals are low in an amino acid called lysine, but
high in methionine and cystine. Lima beans, soybeans and kidney beans
are high in lysine but low in methionine and cystine. Many cultures,
including Mexican and Indian cultures, have limited animal protein
sources but eat combinations of incomplete foods. Examples of
appropriate combinations include:

rice and beans 
cereal and milk 
beans and corn 
bread and cheese 

Recent research indicates that such combinations need not be eaten at
the same meal. If they are consumed over the period of a day, the
necessary building of muscle and body tissue will occur. Vegetarians
thrive on non-animal protein diets because of our body's ability to do
this."

Daawat: Food Nutrients - Proteins
http://www.daawat.com/resources/feature/nutrition/proteins.htm

"But people shouldn't worry too much about incomplete proteins.
Different plants vary in the type and number of amino acids they have,
so people can get the proper amount of these substances as long as
they consume adequate amounts of protein from a variety of foods, even
if meat is not one of them."

Harvard Health Letter: The Neglected, Nourishing Bean
http://www.findarticles.com/p/articles/mi_m1585/is_11_23/ai_50338511

Google search strategy:

Google Web Search: "incomplete protein OR proteins" nourishing
://www.google.com/search?hl=en&q=%22incomplete+protein+OR+proteins%22+nourishing

I hope this helps. If anything is unclear or incomplete, please
request clarification; I'll gladly offer further assistance before you
rate my answer.

Best regards,
pinkfreud

Answered the question well - main source is fairly old by rapidly
changing health standards, but mostly likely still valid.  Provided
additional information, making an interesting and well-rounded answer.

Try reading the Sports Supplement Review 3rd edition, by Bill Phillips
He's the one who does the "Body for Life" program. I read and it's
full of comparisons you're looking for.