You could use tcpdump or wireshark to monitor network traffic to and from that MAC address.
I tend to use tcpdump to do the packet logging and (if i need something a bit nicer than just reading the tcpdump output) i save the packets to a file and load them into the GUI version of wireshark for analysis.
For example:
MAC='01:02:03:04:05:06'
tcpdump -l -n -i br0 "ether src $MAC or ether dst $MAC"
You can tweak the filter rule given to tcpdump to exclude stuff you're not interested in (e.g. port 80 or 443) and/or include stuff you are interested in (e.g. tcp and udp ports 137-139). Don't exclude too much, though, because you never know what kinds of packets will give you the identifying info you are after.
To write the packet log to a file for analysis with wireshark, use tcpdump's -w filename option AND its -s snaplen option (with snaplen of zero to capture full packets)
tcpdump -l -n -i br0 -w macdump.log -s 0 "ether src $MAC or ether dst $MAC"
Leave it running long enough to get a good sample - which might take hours if the target machine is switched off or inactive.