Near Rhymes:
* singer
* dinger (slang for a hit)
* ringer (a person who is very similar to someone else)
* finger
* bringer
Eye Rhymes:
* wringer (a device for squeezing out water)
Slang or Informal Rhymes:
* binger (someone who binges on something, like TV shows)
* stinger (a sharp, biting remark)
Keep in mind that "whinger" is not a very common word, so finding rhymes that are also commonly used can be tricky.