I'm assuming that by democracy you're also implying a republic. While both seem very unlikely, I can think of two ways this could happen.

The first is for a modern stable and successful democracy with a historical monarchy to have some form of popular support for the ceremonial reinstatement of the descendants of the former monarchical dynasty. If the country is doing well under the democratic government and most people feel included in the governmental process, there won't be much support for changing the way things actually run, but they could hypothetically desire the romantic image and cultural prestige of hosting nobility. Within Europe, whether or not a country still has a monarch has little impact upon the function of their governments, so while I still think it very unlikely, its hypothetically possible that, say, Portugal comes to admire the Spanish king enough to want to restore their own historical monarchy. This would also be helped if the monarchy did not end in a way that reflected poorly upon the nation or was intrinsic to the founding of the modern incarnation of that state. So I would think a state like Portugal, who's monarchy was ended by the Estado Novo regime that the present day democracy overthrew, would have a far better view of the return of their monarchs than say, Germany, who's regime fell to popular uprising and in shame, and has since been predominantly tied to far right nationalist movements that the present German state sets itself against.

The other situation I can envision is basically the opposite of that. A country that, while still a largely free democracy, is beset by some major societal problem, be it a bad economy, governmental corruption, war, etc. and comes to view the days of a former monarchy as better times. Of course, if things are bad enough people are likely to lose faith in their government entirely and support its total overthrow, but an intermediate situation, where government still largely serves the people but pervasive issues remain, could prompt the desire for a symbolic return to when things were better as a way to turn a new leaf in the country. Places like much of Latin America and Southeastern Europe are what I'd imagine here, where life isn't necessarily great but isn't entirely intolerable either.