Often I come across phrases like "We believe theorem $X$ is of independent interest"... I know this a phrase specific to math since I can google the phrase "of independent interest" and get mostly math papers.
I think this phrase is rather strange since it leaves out who it might be interesting to or why. Or maybe it is self-evident. Finally, when is it okay to include such a result?
I can provide specific examples if necessary, but it doesn't take long to stumble across in one's own reading.
On
It's not a technical term -- just a conventional way of saying:
We think some people would find theorem X interesting even if they don't care about the particular problem we're using it to solve in this paper. Therefore if you're trying to judge how important our work is, please don't consider only the main problem in the title, but also which other uses Theorem X might have.
Sometimes Theorem A is very technical, and really of interest only to help you prove Corollary B. But sometimes Theorem A might be be interesting to other people to prove other results, unrelated to Corollary B.
For example, maybe Theorem A is "two plus three is between four and six", which is used to prove Corollary B "two plus three is five". On the other hand, if Theorem A is "an even number plus an odd number is odd", and Corollary B is "two plus three is odd", then Theorem A is of independent interest.