recently got a phone from my cousin and her Apple ID keeps popping up asking me to sign in her password

Log out of her account(s) by tapping on her id in the relevant sections (Settings > iTunes & App Store ; Settings > iCloud ; Settings > FaceTime ; Settings > Messages > Send & Receive), and reset the phone back to factory defaults to remove all of her apps etc. (Settings > General > Reset > Erase All Content & Settings) : What to do before you sell or give away your iPhone, iPad, or iPod touch - Apple Support

When and where it is poping up?If there are apps downloade with other apple id then in the itunes and appstore is signed in.it is enough to delete those apps whitch are not downloaded with the signed in apple id

They aren't your apps, if you are being asked for her id her account downloaded them