What's the meaning of "on somebody's account"?

"On someone's account" means "on behalf of that someone". Generally, it is used to indicate that the person shouldn't go to any trouble on behalf of the speaker:

Don't change your plans on my account.

Or it may be used to stress that the person should act for their own reasons or under their own judgement:

Don't change your plans on Fred's account.

The difference is subtle and imprecise.

It means "please don't change your plans for me."

It can have the connotation that the speaker may not wish to accept a favor from you, or it can simply mean he or she doesn't want you to go to any trouble on his/her behalf.