Suffrage is the right to vote in public elections. Universal suffrage means everyone gets to vote, as opposed to only men, or property holders. Suffrage has nothing to do with "suffering," unless the wrong person is elected.
Female supporters of women's suffrage in 1906 were called suffragettes, because the French suffix –ette was trendy back then. But nowadays words with -ette are shunned because they imply the inferiority of women. So it seems that female supporters of a woman's right to vote were referred to by a sexist term.