Will changing server_id on master break replication?
It should not break replication provided you do not choose a bad server_id. Here is why:

Given this server_id setup


Master with server_id 1
Slave with server_id 2


Here is what happens with Replication


Master completes and SQL command
Master records its server_id along with SQL
Slave reads Binary Logs of Master and stores them in Relay Logs
Slave reads reads Relay Logs and sees server_id 1
Since the Master server_id and Slave server_id do not match, Slaves run the SQL


What happens if you set the Master's server_id to 2 ?


Master completes and SQL command
Master records its server_id along with SQL
Slave reads Binary Logs of Master and stores them in Relay Logs
Slave reads reads Relay Logs and sees server_id 1
Since the Master server_id and Slave server_id match, Slaves does not run the SQL


The only bad server_id to assign is the same server_id as the Slave. Other than that, Replication should not break.