Ink stains are miserable to remove from any surface; on leather, they can simply ruin a piece. The good news is that imitation leather is not leather. It is a synthetic cloth and hard to dye, so it can be cleaned in ways that leather cannot. Ballpoint ink is filled with thickeners, sometimes rubber, so does not have the penetrating power of fountain pen ink. A quick survey of dry cleaners and of Dennis McCarthy of Professional Leather Care, of Columbus, Ohio, reveal some strategies.
Hand wash the spot in warm water with a mild soap like Ivory or Dove. These soaps (and their detergents) are hard on real leather; but imitation leather is made of plastic material, and so can withstand this treatment. Use your fingertips, rather than a cloth, to rub the spot.
Pat and dab at the stain and rub gently with a soft cloth (like cheese cloth, or a worn out T-shirt); do not rub vigorously. Synthetic materials have only a veneer of authentic-looking material. If you rub away that veneer to reveal the foam-like inner layer, the item cannot be restored.
Wash the item (if it is small enough) in a commercial washing machine. Use cold water on a gentle cycle with ordinary detergents. This may not remove the stain in its entirety, but because synthetic materials accept dyes poorly, the stain will be far less noticeable over time.
Try the spray-on/brush-off stain removers for faux suede. These products from companies like Kiwi typically come in a spray can with a bristle brush on the cap. Remember, you are dealing with a synthetic cloth, not leather, and cloth treatments will be more successful.
Take the item to a dry cleaner if the above does not work. A dry cleaner can pretreat the spot with common solvents, which should not damage the item.
Seek a professional leather worker to match and paint the item if the item is valuable enough. He may not be able to remove the stain but can cover it with readily-available paints.
Faux suede appears delicate but can be washed like a woollen item and will not shrink. Wash it in cold water with a mild detergent such as Woolite, and hang it to dry.
Do not scrub your faux leather item. If it is a smooth finish, you will remove the finish and leave a rough spot. If it is a faux suede, you will remove the nap.