Yes, it is possible to send a personalized message along with your invitation in LinkedIn. For you to send a personalized message in your invitation you should be connected to the latest iOS and Android versions of the LinkedIn mobile app. After you have updated your LinkedIn app to the latest version you will be able to personalize the invitation. When you send invitations from the People You May Know feature on your mobile app then you would not be able to send personalized message.
Sending invitation message from LinkedIn Android app:
Open the LinkedIn app on your mobile app.
Then tap on the Menu icon found in the top right corner of the member's profile.
Now, tap on Personalize Invite.
Type in your personalized invitation message.
Then find Send on the top right and tap on it.