You can expect a textural change upon thawing frozen papaya, but if you follow the three commandments of freezing food -- freeze individually, freeze quickly and freeze thoroughly -- you'll minimize it. The larger the pieces of papaya the longer it takes them to freeze, and the longer it takes them to freeze, the more ice crystals that form within the cell walls. Ice crystals within the cell walls of fruit and vegetables are responsible for the mushy texture you see after defrosting; by freezing small pieces individually, you'll get a texture as close to fresh as possible after defrosting.
Peel the skin from the papaya and slice it in half lengthwise. Scoop out the seeds using a spoon.
Slice the peeled and seeded papaya into 1-inch pieces. Place the sliced papaya on a sheet pan or tray, spacing each piece at least 1/4 inch apart from the next.
Place the tray in the freezer. Freeze the papaya until solid throughout, about 45 minutes. Remove the pan from the freezer.
Pack the frozen papaya into heavy-duty freezer bags and mark them with the date. Return the bags to the freezer and store up to one year for best results.