The 18-year-old, who plays in a central role, initially joined the Ibrox club on an academy deal but has impressed new boss Mark Warburton in training.
The former Manchester United academy player, who left Old Trafford earlier in the summer, has now earned a full-time contract which ties him to the club until at least 2017.
Thompson, who was born in Northern Ireland, told Rangers TV: “It’s been brilliant so far. For the first few days I was training with the U20s, and then I was training with the first team.
“It was a brilliant feeling signing my contract – I’ve been over for a match before with my dad, but today it just amazing to sign.
“I’m very creative within the middle of the park. I can tackle and make a good pass, and hopefully I can show fans that over the next few seasons.”