pay attention to what they're doing. if it looks like they're doing homework or reading a text book, they're probably studying while they're eating. i've done that before. but if they don't have a phone, and just staring around at other people looking lonely, then go up and say hi. ask them their name. what grade they're in. plans after high school. if they like sports, etc. start up a conversation. then if it goes well, you can ask them to join you tomorrow.
you seem like you answered your own question, and knows what it's like to be alone. so go out, smile and say "hi, can i join you?", and see what happens.