A few options:
>Let soldiers decide
Whoever can sway the preexisting military to their side takes the throne. No need for bloodshed unless the army becomes evenly divided. The monarch's expected to take charge of the army anyway.
>Elective monarchy
Like a presidential election, except electing president-for-life. No political parties, everyone an independent. Could be a clusterfuck.
>Electoral College
A body of qualified candidates chooses one of their own, like the College of Cardinals for Pope. Though that just shifts the question to who gets to be one of the cardinals…
>Existing Claim
Get geneaologists and historians to track down whoever has the faintest claim to legitimate rule over the territory you're talking about. Might turn out to be some nobody (or possibly worse, someone already king of somewhere else), but it's not that crazy, this has actually happened in the real world when nations decided they wanted a king (or wanted to change the one they had…)
>Random Selection
If you really believe the ones best suited to power don't seek it, or that people thrust into greatness tend to live up to it, this might not be so bad. For extra fun don't exclude criminals and the mentally ill.