Mio sounds fine in this case, but finisco tutto il pranzo would work, too.
Often it's just a matter of style and one's way of speaking. At times you need to use possessives, as in Questo è il mio libro, to state without ambiguity whose an item is. Other times, as in Porta il cappello / Porta il tuo cappello, they can be omitted because the property is already implied. In case they aren't left out, they may sound redundant or they may intensify a statement. Porta il tuo cappello sounds stronger to me because Porta un/il cappello is what one's usually used to hear.
The same rules can be applied to personal pronouns, too. Io/tu/lui/noi... are left out most of the times, and when they are used, it's usually done on purpose, as in the early stages of a language course, or to clearly specify the subjects. Eg. Io lavo la macchina, tu lavi i piatti.
Here's another case where it's [way] better to omit the possessives:
Non mettere il cellulare in tasca! Don't put your phone in your pocket!
With possessives, it would be Non mettere il tuo cellulare nella tua tasca!
Grammatically perfect, but sounding like fingernails on a chalkboard.
Practice makes perfect. Keep practicing and the use of pronouns will come natural!