Clearly certain notes will make improvisation easier than others for most people, although a well trained musician could make something interesting out of any combination of notes. A great number of tunes (virtually countless) are comprised of the notes of the major scale, particularly C,D,E and G, for example.
The reason for that is that those notes are related to one another in ways that facilitate building melodies from them: They readily form natural sounding, consonant melodies. So it would also be easier to improvise a tune from them that sounded "good" to most people's ears. (Arguably, every tune originates as someone's improvisation on a certain group of notes, although usually more than just 4.)
On the other hand, if one were to use for example C,C#,F and B, improvising a pleasing, coherent tune would be more challenging, since those notes relate to one another in more dissonant, 'difficult' ways.
In the end, I suppose it's all relative: An accomplished jazz musician, or for that matter a very talented but untrained person could/would make something interesting out of anything. It all depends on who is trying to make the tune, how much time is given, and what's called an acceptable result.
So I'll qualify my answer: Deem it applicable to an "average" person with little or no musical training and 5 minutes of time to produce something.
I've just watched this cute video of a girl improvising a piano song
based on a 4 notes which are taken randomly from a hat.
It's not really a new idea, although that format is "cute". Some jazz musicians have been doing essentially that for quite some time. Legend has it that the great jazz pianist and composer Thelonious Monk, in order to challenge himself, would sometimes build a solo by playing a few random notes, and then connecting them together into a coherent theme that fit in with the tune he was playing. Similar stories have also been told about Charlie Parker.
