This answer has four parts, so here’s the TL;DR:
(1) you don’t need something better than lithium
(2) …it depends on the application
(3) “li-ion” is a broad term and we’ve already moved on past the original commercial cells. It already isn’t what it used to mean.
(4) I suppose you mean literally the same mechanisms of action, but a different ion?
(1) This question fits neatly in the subset of hopeful questions suggesting an upcoming breakthrough. That a new technology will replace what we have and swoop in, deus ex machina style, from the clutches of societal collapse due to climate catastrophe. The reality right now is that we have sufficient technological tools to resolve the problems we face. We just haven’t agreed, as a society, that fixing the problem is a greater priority than our current fictions of economy, central government, value and law. Please stop asking for something better, the engineers and scientists (I’m technically both?) are getting tired of our work not being applied.
(2)…It depends on the application.
Lead acid still dominates automotive starters, due to price. Energy and power are not as closely coupled in flow-cells, which makes them a neater fit for utility scale projects. Li-S set the record for the longest autonomous solar powered drone, due to its lower weight.
Li-ion is scaling up fast and has decent enough power and energy density that it’s a good fit for so many applications, it appears to be taking over.
(3) To be clear though, li-ion is a broad term that loosely means the following combination: graphite anode, lithium metal oxide cathode (LiCoO2, LiMnO2, LiMn2O4, LiFePO4, NCA, NMC etc.), liquid carbonate based electrolyte solvent, LiPF6 electrolyte salt. It’s not really a single technology.
Arguably solid-state cells are on the near horizon, that can match li-ion in performance, and dramatically improve safety. Solid state is a reference to a solid electrolyte.
Silicon anodes have been just over the horizon for a decade. Would Si/NCA still be reasonably considered Li-ion when the original Li-ion were Gr/LCO?
(4) Lithium as the shuttle ion is great and there’s little engineering cause to replace it.
Politically it might ease global tensions to switch to sodium, calcium or potassium. As they are meaningfully more abundant. But those cells hold less energy, have lower power and are no where near as well developed and as such do not cycle as long as li-ion cells.
Lithium has the best voltage window vs ion mass, which means in theory, it can store the largest energy per unit mass.
Strontium has an even greater voltage window, but is much heavier. Calcium has an almost identical voltage window, but is also meaningfully heavier and far more reactive. Potassium is dangersouly reactive and has a smaller voltage window. Most anything else relevant has a smaller voltage window than lithium and are all heavier. It’s not going to get better than lithium.