Sidious/Palpatine was "the phantom menace," that much was clear long ago. Palpatine takes advantage of a major crisis and maneuvers to get elected Supreme Chancellor....Maul (the Sith apprentice) is defeated and believed dead....and as Yoda and Mace discuss which Sith was actually the Master there's a partial camera zoom on the profile of Palpatine as he sternly watches Qui-Gon's funeral pyre.
Palpatine was behind the events in Episode I (of course it didn't hurt that the voice, lower face and chin of Darth Sidious were identical to Palpatine's
). Plus, if you collected Star Wars figures before 1999 you knew that the Emperor's name was Palpatine. So....there's that.
If you parse "The Phantom Menace" into the form "article/adjective/noun", then it becomes synonymous with "The Fake Threat", because "phantom"—when it is an adjective—means "fake", cf. a phantom punch
in boxing. As the fake threat,
the title is clearly referring to the events which were manipulated by Palpatine to provide the pretext for his seizing of power, namely the invasion of Naboo. Now, the threat was real to the Naboo, but it was fake in that the person behind the organization of that threat coincided with the person who rescued the Naboo from it, fake in the sense that the person behind it never intended to carry the plan, as it was understood by the Trade Federation, all the way through. Betrayal of the Trade Federation was always part of Palpatine's master plan.
Now, that having been said, I strongly suspect that in addition to that meaning, the title alludes to a phantom as a noun, which would be an unreal form. I agree that the title plays off the meaning that it would have if you read it as "The Menacing Phantom".
Palpatine is two-faced, so he has elements that are ghostlike. Additionally, Darth Sidious is only seen by the Trade Federation, and first seen by us, as a hologram, another kind of fake form.
Additionally, Obi-Wan's first words when he boards the Federation ship are that he senses something elusive. Whatever he sensed was menacing and ghostlike. That thing that Obi-Wan felt could simply be the Dark Side of the Force itself.
The title could allude to all these things on multiple layers, but the primary
grammatical meaning is "The Fake Threat".