I have an Arduino Uno which is connected to a Nokia 5110 84x48 graphic display via SPI. I want to configure it as a dumb display to display text strings from a PIC, sent to the Arduino over SPI.
That would mean that the Arduino would be SPI master and slave at the same time. The hardware is able to be both, master and slave but not concurrently. Because implementing an SPI slave in software is much more complex than the master side I'd suggest that you drive the display with a software emulation while using the hardware to communicate with the PIC.
Will I need a third data line for slave select or can I hard wire this as it will always be in slave mode?
The slave select line is not for switching between slave and master but it signals the start of the communication. Without that signal the synchronization between the master and the slave will get quite complicated. So I would strongly suggest to use an SS signal.
The data is in the format 8-N-1
Are you sure that this is an SPI interface? 8N1 is the most common format for an asynchronous serial communication (UART) and I never heard of it's use in SPI communication.
Is there a problem because both the display and the PIC are operating over SPI?
Yes, as explained above.