Some ideas:
1. Trace where the potential TX/RX pins go back to the main processor chip and look up their function in the processor's datasheet.
2. It is likely the device emits serial traffic when it powers up. You can probe a potential TX pin with a scope or logic analyzer when it boots to see if there is an identifiable signal.
3. Same as in #2 but try to identify the device's TX pin by connecting only the RX pin of your Bus Pirate to it. You'll have to guess at the baud rate.
4. This is more advanced, but the TX pin should be low impedance and the RX pin high impedance. By connecting a high resistance voltage divider to the pin you should be able to get an idea of the pin's impedance by seeing how connecting the pin to the divider affects the divider's voltage.
1. The router's guts are mostly encased in cans. As far as I can tell, those cans appear to be soldered to the board and it would be a huge amount of work to remove them. Or am I just interpreting it wrong. Companies do solder cans to their boards sometimes, don't they?
2. I don't have a logic analyzer. But I do have a scope. I'd REALLY rather not have to unpack the scope (it's deep in a pile of other stuff in a jampacked cupboard) and try hooking it up (which I'd probably do wrong). I tend to get confused sometimes and connect things wrong, and not a in a good way, hence my reluctance to connect the scope. Maybe if other things don't work.
3. This seems more doable for me. I'm still not clear, however, which pin is Rx on Bus Pirate and which would be Tx on the router. Can you help with that?
Ah, is this it?
https://learn.sparkfun.com/tutorials/bus-pirate-v36a-hookup-guideShould I be using the pins labelled MOSI and MISO on the Bus Pirate?
Master (data) out, Slave in (TX for UART)
Master (data) in, Slave out (RX for UART)
If so, does that mean MOSI is Tx on the Bus Pirate (and goes to Rx on the UART on the router,) and MISO is RX on the Bus Pirate? I ask because I read that some companies like to label pinouts with what they should connect to, not what they do (you know, just for fun).
And all this time I thought Miso was a kind of soup.