I used the following settings to make free gmail account working for me in Database Mail.
Outgoing mail server (SMTP)
Email address: gmailID@gmail.com
Display name: Name you like others to know that email came from.
Reply e-mail: can be same as Email mentioned above or some other email id.
Server name: smtp.gmail.com
Port No: 587
SMTP Authentication: -
Select Basic Authentication
User name: GMailID
Password: Your GMail ID password
Confirm password: Same as above
Save the above settings and hopefully it will work for you as it worked for me.