Charles XII, while campaigning against Peter the Great and Augustus the Strong, deposed Augustus and named his own candidate, Stanislaw Leszczynski, to the throne of Poland-Lithuania. Leszczynski's fortunes promptly evaporated after Poltava, and Augustus returned, with Russian support.
But Peter did not actually want Augustus back on the throne after he was first deposed. When he was deposed, Peter asked Eugene of Savoy to take the throne in 1704. Eugene deferred the decision to the newly-crowned Emperor Joseph, who, having previously been intimidated by Charles XII into permitting Lutheran worship in Silesia, said that Eugene had important military campaigns to attend to and to ask him again next year.
Peter turned to Jakub Sobieski, who turned down the offer, and then to Francis Rakoczi of Hungary, who accepted on the condition that the Sejm endorsed him (Peter never got to ask them, as Charles XII's Russian campaign began immediately thereafter, and after Poltava Augustus marched back in).
How would each of these three candidates have fared as King of Poland-Lithuania if they had openly accepted the throne at the start of the invasion of Russia and fought on Peter's side against Charles XII and Stanislaw Leszczynski? Would their alliance with Peter have deterred Augustus from making another attempt at Poland? Would they be more or less competent than the OTL Saxons? Presumably, Peter would be very well disposed toward them for joining his side at his lowest point, so Russian support against Stanislaw or even Augustus would be forthcoming.