Thanks to Ray Montgomery, Katie, Jozsef Fabian and Barbara Sajkowska
who helped me solve the problem. The final solution follows:

1) Open the polygon feature attribute table (FAT) and add a numeric field to
it
   (for example, name it NewID).

2) Populate the field with unique numbers by highlighting the field name,
then
   opening the field calculator and typing the following expression into the
   field calculator:
   rec+1

3) Stop editing the table and save the edits.

4) Highlight the field title Shape.

5) Open the point FAT, highlight the field title Shape.

6) Click the Join button (or go to menu Table then choose option Join).

7) Hide unwanted fields--leave the field you want to eventually add to the
    polygon active as well as the incremented numeric field (NewID from
    above).

8) Export the table to dBase format. Remember where you put it. Close
    the point FAT.

9) Make the project window active. Click on the Tables item. Add the
    new table.

10) Highlight the NewID field.

11) Open the polygon FAT. Click on the field title NewID.

12) Join the two tables.

Now your relevant point attributes are joined and available for queries
in the polygon FAT.

Jeff Hamilton