What argument(s) for HJ do you consider strong?
It would take too long. I'm not sure I have the enthusiasm for a prolonged debate. I kind of overdosed on this topic at one time. There isn't one strand which tips the balance for me. But if you want an example, I do think that what's in the Epistles may be considered reasonably good evidence in favour of his existence, all things considered. I don't tend to put much stock in the Gospels as sources.
In a nutshell, whatever else about the Epistles, the writer (who, it seems likely, wrote at least some of the letters) was talking about someone. That it was a Judean Jewish bloke who had not long before died seems to me slightly the better explanation than any other. And 25 years after events would be practically nothing in terms of sources when it comes to ancient history. And when trying to determine existence, we are after all dealing with ancient history.
More importantly, what there isn't, imo, is enough strength in the counter-explanations, and imo they are less parsimonious, and sometimes imo just implausible. Earl Doherty's go into this category as far as I am concerned. This view was reinforced after I participated in a discussion with him on an online forum a few years ago. Quite apart from finding him to be borderline crank personality-wise (just my personal opinion) his analysis and conclusions are imo unreliable, dubious, and not very informed, and furthermore I think it was bad judgement on Richard Carrier's part to endorse his ideas. The apparent absence of mythicists at or near the time period in question doesn't help the mythicist case either.
It may be annoying, but as I understand it, in the study of ancient history, allowing probable existence is sometimes akin to the default where there's evidence for existence and unless there's a good case to the contrary.
So in that sense, there's the argument that it's inconsistent to declare him not to have existed when, although it's a close call to make, and agnosticism/neutrality or something close to it (in either direction) is/are arguably the most warranted positions, there's arguably as much evidence for him as we have any reasonable right to expect in the circumstances, and possibly more evidence than for many other figures of similar status or renown from ancient history, during their lifetime I mean. Name me
any messianic claimant, preacher or magic man from 1st Century Judea, for instance. There were quite a few, apparently. Then, tell me how we come to know about them. At least one, apparently, had more followers than Jesus is supposed to have had at the time.