Though most children look forward eagerly to summer, there comes a point after a stretch of idle weeks when a common lament becomes:

"What can I do now?"

To relieve the summer doldrums, you may wish to try one or more of the following activities.

1. Help your children trace their roots. Most libraries have books on genealogy, but you don't need one to begin a simple family tree. Provide a notebook that is divided into sections for maternal and paternal families. Give them as much information as you can about your parents, grandparents and great-grandparents. Include full names and dates and places of births.

When your own memories and files are exhausted, provide the names and addresses of other family members to whom your child can write for more information. Older relatives, in particular, enjoy helping children on such family projects and are pleased to relate favorite family anecdotes and stories. This activity is appropriate for siblings to work on together.

2. Introduce them to hobbies. The post office has booklets on stamp collecting, and many hobby shops have inexpensive paperbacks for beginners on stamp and coin collecting. If your child shows an interest in collecting, summer yard sales often feature items children like to collect, such as dolls, bottles, license plates and baseball cards. Antique buffs are often mines of information concerning collecting of various kinds.

3. Encourage them to construct an elaborate town or village with available toys. This is a marvelous project for children from 4 to 10 or even 12. They can use every block, Lincoln Log, Tinker Toy and Lego in the house, as well as road and train sets and anything else they can think of to enhance their layout.

An essential requirement is sufficient space to allow the construction to go on for a period of days. (Nothing is more defeating than building something wonderful and then having to dismantle it before suppertime.) A basement or an unused bedroom is ideal. When completed, the village can be used for playing games with plastic or wooden figures. Take a photograph of the whole thing for posterity.

4. Help your children begin a botanical journal. Purchase or borrow from the library identification manuals on trees and plants in your area. Buy a sketch-pad, or make a homemade journal by stapling unlined paper together with a suitable cover of cardboard or construction paper. A good set of colored pencils is a great help with this project.

The child tries to identify every plant and tree on your property, recording both common and species names in a journal, and drawing a representative sketch of each. If you live in an apartment, or if your own property is too small for this type of project, you might wish to ask nearby neighbors for permission to include adjacent territory. Parks are also suitable places for this type of activity.

Before beginning this botanical project, check with your extension agent regarding poisonous plants, and be sure your child knows how to identify them.

5. Join them in getting to know a part of the world. Select a single country or a small geographic area of the world with your child, perhaps the country of origin of your own family. Help your son or daughter make a large-sized map of the selected area on poster paper. Post it near your dining room or kitchen table.

From the library, select appropriate books about the country on a reading level suitable for the child. Do not overlook fictional work, which often provides more detail than textual material (and also encourages reading). Each evening at mealtime, you and your child each provide three interesting facts about the area to the rest of the family. Some children balk at geographic projects, so in the beginning stages try to find at least one zany or intriguing fact about the country to whet the interest. You might wish to include in your studies the preparation of ethnic foods and attendance at a local celebration having to do with the chosen country.

6. Help your children build a fort. It's a summer standby. Appliance boxes make excellent walls, especially if you can procure several to attach together. Supervise or help with the cutting away of doors and windows. These marvelous hideouts can be painted, written on with chalk or crayon, curtained, and otherwise embellished at no expense. More substantial structures can be built of 2x4s and plywood, moving crates or other scrap wood.

7. Or suggest they create a puppet theater. Appliance boxes are handy here, too, providing room inside for the puppeteers. The "stage" can be a cut-away portion of the box. The library has books on puppetry, as well as books of short plays that can be used. Many children prefer to use familiar stories such as "Goldilocks" and "Red Riding Hood." Puppets can be made using an amazing variety of materials ranging from old socks and pipe cleaners to elaborate papier-ma che' constructions.

8. Encourage your children and their friends to write and produce their own play. If done correctly, this activity can involve many weeks. Begin by providing them with short plays from the library, having them read roles in several of them to get the feel of drama. Then they can select their own theme, plot and characters. Sets and costumes can take an enormous amount of time and provide a creative outlet for children of all ages. A particularly valuable aspect of this activity is that it enables children of different ages to work cooperatively together. Invite parents and friends to the production, and provide refreshments.

9. Prepare a make-it box. Place it in an easily accessible spot. As its name implies, this is a box containing materials to make things. Include various kinds of paper, empty cereal boxes, plenty of tape and glue, string, rubber bands, seals, ribbons, sewing scraps and scissors.

10. Help your children begin a scrapbook. There are many different kinds of scrapbooks. Some are used for keeping mementos of one's personal life; others are devoted to a particular subject. One especially lovely scrapbook in our family was a gift to our daughter from an elderly aunt. In it, Aunt Lovina had written in her own hand a number of pieces she had been required to memorize as a child in school over 70 years ago. She had painstakingly looked through magazines to find suitable pictures to go with each, and these were pasted in the book beside each piece of prose or poetry.

11. Organize an art, sewing or craft project. This is easier than it sounds. If you have several children in your neighborhood who are approximately the same age and share an interest in one of these areas, you might consider hiring a talented teen-ager to teach them.

12. Help your children prepare a wildlife garden. Most children enjoy this project enormously for two reasons: Almost all of them like observing wildlife, especially if it is done close-up, and the results are seen almost immediately. Gardening With Wildlife, an outstanding book on this subject published by the National Wildlife Federation, is available in book stores and in many libraries. Profusely illustrated, it gives clear directions for constructing feeders of various kinds. It also decribes the types of plants that attract various species and the kinds of food to place about in your yard. Seldom does it take more than a day for birds to discover a pie-pan birdbath, and in less than a week, your child can be recording which birds eat the various kinds of food placed on the feeders.

13. Initiate a room redecoration project. This can be both long-lasting and productive for children 10 and over. Provide water-soluble paint, brushes and clean-up materials, but allow the actual work to be done by the child as much as possible.

In selecting paints and fabrics, children learn a great deal about color, texture and harmony. Furniture can be treated in any number of ways, and don't neglect the advantages of decals and stencils in decorating rooms. Your child may wish to paint, instead, a mural on one wall. Encourage the child's creativity by helping prepare a drawing on the wall to be painted, using a grid and a plumb bob for measuring. Dry tempera paints mixed with nontoxic white glue make an excellent paint for this project.

If you succeed in perking up a bored youngster with one of these projects, you may spark an interest that will last a lifetime.