What does it mean exactly " averaging over eight symbol periods" when calculating RSS?
A "symbol" is a signal pattern that is used to encode a particular value for transmission.  For example, in the world of asynchronous binary transmission, sometimes referred to as V.24, async, or RS232 (the last of which is incorrect, but it's still widely used) a symbol is a start bit; between 5 and 8 data bits, LSB first; an (optional) parity bit; and one or more stop bits.  In its (arguably) most widely used form, a 10-bit symbol (1 start, 8 data, 1 stop) can be used to encode and transmit any value in the range 0-255.

A "symbol period" is the time taken to transmit a symbol.  Referring back to my previous example, a commonly used transmission speed is 9600 baud, or 9600 bits per second.  In my example there are 10 bits in a symbol, so 9600/10 = 960 symbols can be transmitted per second.  The symbol period is 1/960s, or 1.042 milliseconds.

Transferring this example to a radio communications system, which could use a similar symbol encoding method and symbol rate, averaging the RSSI over 8 symbol periods would mean taking several RSSI measurements in the time taken for 8 symbols to be transmitted (8 * 1.042 = 8.333 milliseconds) and averaging them.

Averaging the readings in this way gives a more realistic indication of the useful signal strength. A single reading might be substantially higher or lower than the average due to random interference, but if you take an average over time then peaks and troughs tend to be smoothed out.  This is actually a simple example of filtering.