といおうか

といおうか means "how shall I put it", "or should I say": you offer a word for something and show that you are not sure it is the right one. It follows a plain-form word or phrase directly; a noun or な-adjective takes no だ. いおう言おう is the volitional of ゆう言う, so the phrase is literally "shall I say".

With なん何 in front, なん何といおうか stands on its own as "how shall I put it", ahead of the description.

AといおうかBといおうか

Two candidate words, each with といおうか, say that the thing sits somewhere between them and neither word fits on its own. The sentence then usually gives up on naming it: とにかく, ふくざつ複雑な, ことば言葉ではあらわしにくい表しにくい.

Register

といおうか belongs to writing and careful speech: essays, speeches, reflective narration. In conversation the same hesitation is というか, and in polite speech といいましょうか. とでもいおうか adds でも and is even more tentative: "I might call it…".