If we're speaking US presidential elections, I think a pretty sensible choice would be the two-round system. It seems much more equitable than the current one, but also simple enough that it can be easily understood by most US voters. (Think of the first round as of primaries, just across party lines.)
It's also quite well tested, having been used all over the world. (It still has downsides. It's a majoritarian system, so vote splitting still applies — there's a slim chance you could end up with an unfavorable pairing in the second round; think Trump vs Cruz if you're Dem-leaning, or Sanders vs Clinton if you're Rep-leaning.)
Recently, I have also been pondering possibilities of pretender-based systems, inspired mainly by the way chess championships are organized: first, a pretender runoff, then a match between the current champion and the pretender.
Imagine something like that:
· some time before the term's end, the president announces their start as an incumbent (or in the case they can't / don't want to run, they appoint a successor) — for instance, Obama would nominate Clinton;
· during the following months, pretender candidates would announce their campaign — for instance, Sanders, Trump, Cruz, Johnson and Stein would run;
· the first round would be a runoff between all the candidates except the incumbent/successor, then the most successful opposition candidate gets to the second round — for instance, Sanders or Trump would win the first round;
· the second round is a vote between the incumbent/successor and the winner of the first round — for instance, Clinton vs Sanders/Trump (whoever got more votes in the first round).
In theory, this kind of a system would encourage more stable governments, making it easier for successful political teams to continue ruling, but also making it easier for voters to vote out an unsuccessful government. Basically, the vote would come down to “continuation” vs “change”. It would also reduce the chaoticity and the influence of vote-splitting (since one of the contenders would be known in advance).
In terms of equitability, it would be somewhere in between the current system and the two-round method: the first round is free for all, making it easier for independent candidates, but there is only one winner of it, making the competition more tight.
There is some room for strategic voting, though. Since the incumbent doesn't take part in the first round, their electorate would be able to vote for other candidates, making room for exploits. Imagine die-hard supporters of the incumbent/successor (say, Clinton) voting for a lesser-known pretender posing less threat (say, Johnson) in order to bring down the top pretenders (say, Sanders or Trump). It involves boosting one of your candidate's competitors, though, which is quite dangerous and can backfire spectacularly, so I don't know how common such a situation would be (it could be anything between ‘not at all’ and ‘basically every time’). Also, in some way, it could already be happening
(That would also involve a rather large number of incumbent/successor-voters boosting the same pretender candidate. A more realistic situation would be something like Clinton supporters splitting their votes: some of them voting for Sanders, some for Johnson, some for Cruz possibly, and then some abstaining.)