Luckily, a friend of mine has a cousin called Nienke! This is what she gave me:
"Can you believe it, my real names are Trijntje Bootsma (my Frisian grandmother's first name and maiden name). But according to my Frisian father the nickname (noemnaam) for Trijntje is Nienke - not sure why, but I know another Frisian Nienke who's real name is also Trijntje. I think it is quite possible that Trijntje comes from Catharina and then became Nienke."
So, there we have it! She's right about Trijntje coming from Catharina - it would be pronounced more-or-less like train-ke - and, I think, confirms my suspicion.