What does my child call my godparents? Do my godparents become "great godparents" now that I have a child, similar to my parents becoming grandparents?
To most people in English "great godparents" means simply godparents who are very very good at it. In general, "godparent" means different things to different people. People especially differ as the the religious connotations. For many Christians, a godparent has promised to assure that the godchild will grow up properly schooled in the faith, if the parents are unable to.
People who want a Chinese term for godparents have used 乾媽 and 乾爹. But I think in China if you say someone is your 乾媽 or 乾爹 most people will think you mean that person is your foster parent or step parent. In the west a godparent might become a foster parent, if the parents die, but usually that does not happen.
By far the most common use of 乾媽 on line is to name 老干妈 chili sauce. And in Taiwan at least one person has used 乾媽 to mean oranges. See http://news.ltn.com.tw/news/society/breakingnews/1593333
So I think if you want to express your idea of godparents in Chinese, you should first be clearer about your idea in English. If you mean, for example the Catholic Church sense of the term, then you could ask Chinese Catholics how they express it.
To answer your question directly, and assuming there is no religious context :
If you refer to your godparents as 乾爹、乾媽，then your kids could technically refer to them as 乾爺爺、乾奶奶，or 乾公公、乾婆婆。Depending on how close you are to your godparents, you could drop the 乾 as long as there's no confusion between references to the great godparents and the blood relations grandparents. Another idea would be to use "last name"+ 爺爺奶奶 or 公公婆婆。
Godparent is translated as 教父母. Since Christianity isn't major part of historical Chinese family life (not now either), there is no predefined way for great-godparent. So it's entirely up to you what you child should call him/her.
The most common way would be to simply call the great-godparent 爷爷奶奶 when directly talking to him/her. Inside family, to specifically differentiate him from genetic great parents, you can add the surname eg 王爷爷 or just make up the term 教爷爷 for your family. Outside family when you child introduces that person to a third party, say it in full 爸爸的教父.