Let's be clear this is all an extreme hypothetical, right? And there are all kind of reasons
not to do this; to be sure - I don't want to claim it's somehow a
good idea to do this; just that in the context of discussion at this point - i.e. given that you retain the current structure of power separation, and the current voting schemes, and decide to call for fresh elections on a failed impeachment,
then that there is a case to be made to reelect the entire senate as opposed to president or house. However, that precondition is an extreme hypothetical.
The reason to favor the senate over the president is clear: if they truly believe the president still should retain political support, the president isn't the you should force into a new election.
The reason to favor the senate over the house (if forced to pick just one by this hypothetical) is that the senate is the odd one out here in electoral support, and to make it harder for them to play political games with this choice; i.e. it's in their interest to pick a stance that is supported by voters, because if they don't they'll answer to them. It's also the most impactful reelection to force, because those elections are otherwise spaced the furthest apart. Essentially: by giving the house more power and the senate less, you're inching closer towards the greater stability of a parliamentary system, and additionally it has the most impact because the senate is the least frequently reelected normally. The house really should have supremacy over both senate and president; and it's OK to check that supremacy with voters to make an unleashed house suffer the consequences (because a house that really did abuse this power is still up for reelection quite quickly, and results from the senate election would be very publicly visible).
To be clear; it's just a hypothetical. It'd be better if the senate were simply dissolved, or relegated to a more advisory role as it is elsewhere, or merged into the house, and better if presidents were elected by congress, not directly - but given the hypothetical of just one releection to force, I think there is an argument to be made that a senate relection would have the most impact on the impeachment process.
To be clear: I don't think the US president being hard to unseat is a feature; that's a design flaw. Other democracies have stable foreign policies to, even when just a simple majority in the equivalent of the house is enough to immediately unseat the equivalent of the president, with no recourse to the senate at all. In fact, other democracies seem to have a more stable foreign and domestic policy, because there's less infighting and less need for hyper-speed policy making in the rare moments without gridlock; none of that "but we can't approve that deal because currently the house or the senate aren't in the hands of the same party as the president". The decision-making capability of a parliamentary government isn't perfect, but it's less likely to gridlock for long periods of time.
It also means there's no hope of any party of trying to blockade the other should they lose, so there's more incentive to seek consensus beforehand because they simply won't be able to force anything should an election ever go the other way. And there's more potential cost for reversing previous governments policies, because what's to stop a future government from playing that same game? As it is currently, a party can risk being quite confrontational, knowing that a trifecta isn't all that likely in general. There's a good chance any controversial policy made will stick around simply because there's a good chance future governments will be gridlocked for a good while; worth risking that gamble. If governments knew that a future government could reliably overturn any policy they made, they'd be more careful choosing partisan pet projects, especially those without clear popular support.