There would be tradeoffs depending on what you want. A Hohmann orbit would be the minimum energy needed to reach Jupiter, but you can only access it during a brief window that opens up when Earth and Jupiter are in the right relative positions, which would happen at intervals a bit longer than an Earth year. (Earth returns to the same place it was in at the last window in one year, but by then Jupiter has moved on. A Jovian year is a bit under 12 Terran years so I guess the window is about every 13 months. Also since both planets are in slightly elliptical orbits, some windows are more advantageous than others (and their timing won't be at a constant interval either).
There might be some advantage launching around the Moon, but the Moon has to be in the right relative position which would often not be the case when the window opens; also there's the "Oberth Effect" to consider--it's better to apply delta-V low in a gravity well because it results in more energy accruing to the spacecraft, Earth's well is much deeper than Luna's, so it's a tradeoff between possibly gaining something from Luna's orbital speed versus burning for delta-V at a higher potential.
Anyway if you want to get to Jupiter sooner, or launch at a more convenient time, we are looking at higher energy orbits, which cost more delta-V, and also if the goal is to reach Jupiter orbit and stay there instead of a flyby, the more energetic the transfer orbit the more velociy we have to lose at that end. If atmospheric braking at Jupiter is an option, that might be OK, though such maneuvers are going to be at very high speeds compared to familiar Earth reentry speeds, since Jupiter is so massive hence the approach speed will be great.
Jupiter is great for gravity assist orbit changes to other goals though the window for the "Grand Tour" taken by the Voyager probes OTL will be closed by now. But if one can reach Jupiter first and be satisfied with a flyby there, I suppose just about any outer planet can then be reached at just about any time, if one is not insisting on visiting a third and then a fourth and fifth...
So if you can send something to Jupiter I suppose that means you can reach any outer planet as the final goal, and get a free flyby of Jupiter into the bargain, if your craft can take the rather adventurous conditions of a close approach to the big planet with its powerful magnetic field, radiation flux, fast-moving orbital junk, and so on.
One approach might be to design a standard Outer-System launch system that can launch a fixed payload on a Hohmann path to Jupiter, and fine-tweak the close approach to Jove so that a perihelion burn of a final stage carried all that way as part of the standard-mass payload sends it out on the best approach to whatever outer planet is the final goal; burning low gets maximum benefit from available delta-V and one ought to get some angular momentum from the encounter too. So the final payload probe mass will depend on just how much rocket you need at periJove, but I suspect the delta-V required there would typically be in one ballpark so the detailed probe designs would be around one typical mass.
I am not sure a hydrogen-oxygen final stage can store the fuel on the long arc from Earth to Jupiter, so we might be stuck with lower ISP propellants. But those have their advantages offsetting their inefficiency to consider--much more compact, minimal maintenance, in the case of hypergolics easier rocket engine design, in the case of ker-lox (assuming oxygen at least can be kept liquid on the way out) a lower combustion chamber temperature.
I'd think then that designing around being able to reach Jupiter in a standard way would open up the whole outer system for exploratory probes. A standard rocket stack designed to send a standard outer-planet packet to Jupiter at Hohmann optimum window time might also send a smaller payload to Jupiter or even a second, final destination by using some or all of the final Jovian-maneuver rocket mass allocation for final increments on the escape orbit from Earth instead, thus enabling trips to Jupiter at non-window times.
And if your program can support multiple launches of the standard outer system launch vehicles at the same time, or in very fast succession, a whole cluster of different probes can be sent to Jupiter in a salvo, there to scatter to different final destinations.
I've lost track of whether this mission is to be one of the standard ones the Argo series is to be capable of, or if this is another specialized variation. The naming of names matters to you and so I've been trying to track down whether Odysseus's ship had a name; I sure don't recall one.
Besides, his return expedition to Ithaca started out with a dozen ships, all but his being soon destroyed, and his own ship is wrecked too, so perhaps even if someone can turn up the name of his flagship, it wouldn't be the most auspicious!
Maybe I can think of some other mythic track referencing a Europe-relevant culture hero famed for far-flung exploration of the most distant lands?
I believe there was a whole genre of Celtic poetry about far voyages, of which the tale of Saint Brendan's expedition is the most famous, "Imre" or something like that.
But based on my track record, I'm guessing if I do find some name or generic term that thrills me, you'll want some variant of it instead.
Maybe, if these rockets are a distinct variant of the Argo series, these versions can be the ones called "Argosy?" That's not really appropriate because the Argosy name suggests a more bread-and-butter workhorse for everyday use, not far ranging-adventure.
But if you'd started with Argosy for the whole series, these deep-space variants could well be the Argos!
