On this router, show qos map interface <interface> displays "5 minute offered rate" for each map entry. What does this signify?
It shows the average bits per second that matched the map entry over the previous five-minute sampling interval.
If its just matches on the map why does it say offered? 5 minute rate would seem more appropriate.
Because maps can be configured to drop traffic. In these cases you will see both an offer and a drop rate, for example:
5 minute offered rate 160 bits/sec, drop rate 40 bits/sec.
In this case the traffic coming in to the map averaged 160 bits/sec but the map rules dropped on average 40 bits/sec so the traffic passed on averaged 120 bits/sec.
If the map is just classifying traffic such as adding, modifying, or stripping DSCP tags but not configured to drop, then all you'll see is the offered rate.