A tricky point.
If you want the sentence to be formally correct (e.g. in writing), you could say: "Anybody who is going to the concert should put his or her name down [e.g., on the list, etc.]."
If you don't mind it being more informal, you could say: "Anybody who is going to the concert should put their name down."
The verb and the possessive adjective should both be singular, to agree with "anybody." The tricky part is that, informally, the possessive adjective "their" can be used in this context.
Does that help (or just make it more confusing?)
