You can call your trust what ever you like, however it is good practise to make the name descriptive of what it is. For example it is better to use ‘The Jones Family Trust’ rather than just ‘Jones’. This makes it much easier for entities that deal with your trust so they know what it is. Also remember that this name is going to appear on bank accounts, share registers, etc so you do not want to use something vulgar or embarrassing.
Discretionary and Unit trusts are not registered with a governing body like companies or businesses. Therefore it is common for there to more than one trust with the same name. For example if you search for ‘SMITH FAMILY TRUST’ on the Australian Business Register you will find many examples. This is fine, asset management or distributions will never be confused as your Trust’s unique ABN will be used to identify it.
