The ink in most ballpoint pens is a pigment made by suspending particles of carbon black in a solvent, so getting it out of wood is similar in difficulty to removing wood stain. You may have some success using a strong solvent to dissolve it, and bleaching may also work, but in the end, you may have to resort to sandpaper or a scraper. The prerequisite for any of these procedures is removal of the finish. Of course, you might get lucky and find that the ink hasn't penetrated the finish; in that case, removing it should be easy.
Procedure for Ink Removal
Mix an ounce of gentle soap -- such as dish soap -- in a gallon of warm water and scrub the ink patch with a kitchen scrubber. If the ink hasn't penetrated the finish, you may be able to remove some of it while causing minimal damage to the finish.
If you notice an improvement after scrubbing with soap, but you can't get all the ink off, moisten a rag with rubbing alcohol and continue scrubbing. Alcohol won't harm the finish, and it may dissolve the pigment. You can also try mineral spirits or acetone, although you should test these first.
The failure to remove the pigment means either that you need a stronger solvent -- such as lacquer thinner -- which will damage the finish or that the pigment has soaked into the wood. In either case, the best approach is to sand off the finish, using a piece of 120-grit sandpaper. Unless the pigment has spread over a wide area, you should be able to remove the finish quickly by hand sanding. Otherwise, use a palm sander.
Moisten a rag with lacquer thinner and rub the wood vigorously. Combining this procedure with more sanding should take care of the stain. If not, you may need to bleach the wood.
Pour some household chlorine bleach into a spray bottle and spray the affected area; then wait overnight to see if there's any improvement. If so, repeat the procedure until all the color is gone. If not, bleach probably won't work, and your only alternative is to scrape out the stain.
Scrape the bare wood with a pull scraper. Draw the blade over the wood using light pressure to remove as little material as possible, and repeat until all the color is gone. if the color penetrated deeply, you might find that you had to remove enough wood to leave ridges. If so, sand the ridges flat with the sandpaper.
Stain the bare wood with wood stain to restore the original color; then apply a single coat of clear finish. If the finish is polyurethane, you can apply it with a brush; if it's lacquer, spray it from a can. Let the finish dry; then sand it lightly with 220-grit sandpaper and apply another coat.