Dab, do not wipe, with a cloth barely damp with surgical spirit or an alcohol hand gel, with a clean cloth pressed underneath to catch what comes through.
- Put a clean white cloth underneath the fabric. Everything you dissolve has to land somewhere, and without this it lands in the layer below.
- Dab the mark with surgical spirit on a cotton bud or the corner of a cloth. Alcohol hand gel works if you have nothing else.
- Move to a clean part of the cloth constantly, and keep changing the one underneath.
- Rinse with cold water and wash normally.
Do not wipe, and do not soak the area. Ink spreads with the solvent, so a wet cloth dragged across a small biro line makes a large blue smudge. On leather, stop and go to a specialist: solvent takes the finish off along with the ink.
Ballpoint ink is a dye in an oil or glycol base, neither of which dissolves in water. Alcohol dissolves both, which is why it works where a wet cloth does nothing.
Household chemicals can be dangerous in combination. Never mix them, keep the room ventilated, and test anything on a hidden patch first. Full disclaimer.