General:
* Edge
* Border
* Margin
* Flange
* Circumference
* Perimeter
* Outlining
* Boundary
Specific to wheels:
* Wheel edge
* Wheel flange
* Bead seat
* Lip
Other:
* Lip (for a cup or bowl)
* Brim (for a hat)
* Verge (for a cliff)
The best synonym for "rim" will depend on the specific context in which you are using the word.