I have the idea of wielding one of the hold's blades as a thane, and just generally living in the hold and carrying out duties as a thane (attacking bandit forts, doing radiant jobs, etc.) So I'm wondering at what point is the blade that a Jarl gives you is generated by the game. Can I reload over and over again when I'm going to the Jarl's house to receive it so that I get a different one each time? I don't want to end up with some useless battle-axe that does magic damage.

Thanks in advance for any help

The choice of axe versus blade was already made when you got the game. You always get an axe for Eastmarch, a blade for most other places, and Balgruuf always gives an axe, and Vignar a blade. But each of those does come from a leveled list, so material, one/two-handed and enchantment is random.

I believe the item is generated when you start the quest by talking to the Jarl. Most quest-related alias item are, but there are ways to defer it. Maybe someone else knows that detail.

Balgruuf's axe seems to be generated when you enter Drangonsreach for the first time and Laila will always give you an Iron weapon as it's set at the start of the game.

The other Jarls, however, are more... flexible and the potential reward will actually reset after some time. For example, I went to speak with Igmund after buying Vlindrel Hall and he gave me an enchanted Iron battle axe, so I loaded up a save I had before talking to him (still in Understone Keep, but asking him again would just net the same reward) and then went somewhere else. After some time (10 days perhaps) I returned, but this time I got an enchanted Glass war axe. Then I tried something else. I reloaded a save before returning to Markarth, then went to speak with Igmund again and this time I got an Ebony battle axe cabable of dealing 30 frost damage. Tried the same with the other Jarls and it was the case with them as well.

So if you stay away from their hold capital for at least ten days and then make a save, you can keep reloading until you get the reward you want with these Jarls.