Aspirin is o-acetyl-salicylic acid (I may have gotten an y or i wrong there). No cite here, but there’s a notion that some of the stomach problems sometimes associated with aspirin intake are lessened by taking it in a tamponated or basic form. The tampon form would be a mixture of the acid and its sodium salt; the basic form is the sodium salt. Both exist as OTC medication in Spain, I don’t know about the USA.
Why sodium, and not lithium or potasium? Because sodium is lots cheaper, and it’s considered sort of “neutral for anybody who doesn’t have HBP problems”. If anybody was to make a mixture involving the lithium salt, they’d want to be able to claim it’s good for the blues; if they were to involve the potassium salt, they’d want to be able to say it’s good for HBP. But in order to make such claims, they’d need to finance appropiate studies. Nobody is going to use the more expensive version when all it represents is a financial loss.
That’s on top of the solubility issue, which would also be affected by the metal used.