With an EASA PPL, can I invoice my mother's company for airplane rental to fly her to a convention, with no other renumeration?
In the UK, at least, the rules specifically state: 


  Privileges and limitations:
    A holder of a PPL may, for compensation or hire, act as a crewmember of an aircraft in connection with any business or
  employment if—
  
  a. the flight is only incidental to that business or employment; and
  
  b. the aircraft does not carry passengers or property for compensation or hire.


Despite what this may sound like, this means you are not allowed to be paid for flying unless you happen to be an employee and flying for work -- e.g. an archaeologist (with a salary from University X) flying yourself and a colleague to a dig site (while employed at the University of X -- thus flying for compensation [because you are paid, indirectly, to do this] but incidental to the employment [the University of X is not an airline, and the colleague you are with is not a co-pilot or paying passenger]). 

I have always been taught that the safest thing to think is this: As a PPL, you cannot fly for compensation, and each of your passengers at most has to contribute not more than their "fare" share of the cost of the flight. 

By invoicing your mother for the cost of the plane rental you would not be contributing that value in kind, and thus would be in violation of this rule (with rather unpleasant consequences, should you be caught -- ranging from losing your license to a minor criminal conviction). Moreover, the fact that you mentioned the word invoice makes me think that your mother's company would be expecting you to provide a service -- all of these are commercial terms, and alarm bells would be ringing in the mind of any examiner looking at this from the outside, as it starts to really sound like you're providing a service for hire. 

Ignoring for the moment that many organisations have rules about doing just the following with family members, I suggest instead keeping all of the receipts of your legitimate expense for aircraft hire -- Hobbs hour fees, fuel, landing, whatever -- and then getting your mother to claim exactly half their value as a business expense. That at least makes it very explicitly clear that you are paying your fair share of the cost of the actual flight, and highlights the nature of the compensation as covering an expense, not a fare.