It could be any of them: battery, alternator, starter relay. Also could be loose or dirty battery or ground connections. Lights coming on indicate nothing. It takes much more power for a starter to crank than to turn on lights. What type of cables are you using for the jump. $10 cables with 10 gauge wire and sheet metal clamps are virtually worthless unless you allow several minutes for your battery to recharge. They can't carry enough power to actually start an engine. A good set of cables that can start a car has 2 or 4 gauge wire and heavy steel clamps.