Is there a synonym to the phrase "running joke" in any context? I feel like I've heard something like onrunning and I'm not sure if there isn't a connotation of an inside joke, that could serve as a synonym.
A running joke
is a thematic piece of recited humour which continuously recurs over time.
An inside joke
is one that is understood only by people "in the know", but may not recur.
You may be thinking of
An on-going joke
which is a piece of recited humour that continues across an entire story line.
an ongoing gag
is more a vignette than a recited joke.
The difference between a joke and a running joke is that running jokes occur over a span of time – sometimes for days or weeks, sometimes for months or years. They aren't simply told once and everyone moves on. In fact, people often find them funny largely because they remember the last time people referred to the joke.
The difference between a joke and an inside joke is that inside jokes are only funny to people who have certain background information that allows them to understand (and therefore be amused by) the joke.
The two are not the same thing, but there can be overlap, in that an inside joke can also become a running joke when it recurs over a long span of time.
For example, let's say we had an aunt named Henrietta who liked to take care of stray cats. It started with one cat in 2002, then she found two more two years later, and five years after that she took in a fourth kitten. We may adore and respect Henrietta, but, because she has taken in four cats, perhaps we still like to joke around and call her a "crazy cat lady." Maybe, every year during the holidays, we ask, "Is the crazy cat lady coming over, too?" Maybe three first three cats have all died and she only has one cat now, but we still call her the "crazy cat lady" because that's the running joke.
This could be considered a running joke because we've been calling her the crazy cat lady for several years now. But it might also be considered an inside joke if not many people outside of her immediate family knows about her love for cats. If I went into her workplace, for example, and asked, "Is the crazy cat lady here?" I might get a puzzled look from the receptionist because Henrietta's coworkers are not aware of the longstanding running joke in the family.
When a person is a running joke, they've come to be regarded as a fool. Nearly everything they do reinforces this view of them. It's one idiocy after another. Hence "running", as a TV series can be "running" -- episode after episode being aired. When a situation is a running joke, the people who have created it and are perpetuating it are seen as fools. They are not "ha-ha" funny but objects of ongoing ridicule.
An inside joke is one that only a small group of people would understand because it has special private meaning.