Can you change your gamertag for free

No

It does cost 800 Microsoft points.

However, if you have in anyway an offensive name, ask players and friends to report it from your profile. It will then let you change it for free,once you have 15 complaints to your gamerscore ( these cannot be seen however)
. For instance mine was STDS 4 YUR MUM, it got reported and i changed it for free. However if your name isn't in the slightest offensive, the xbox live staff will likely ignore it.