Within the universe of the movies and related to all the characters shown, it was not just a random video, but an intentionally made attempt at artistic stream-of-consciousness and an exposure of emotion.
Greg was supposed to be making a tribute film, something "expected"; like a graduation kind of film that could be played for everyone at school to remember Rachel and how she fit into all their lives and would ultimately be a means of shared catharsis and remembrance for her life cut short.
Madison would periodically cajole and encourage Greg throughout the second half of the movie about making this film for Rachel, as Greg was kind of a completely narcissistic self-righteous self-hating "anarchist auteur". He and Earl would make these absurdist, pun-riddled remixes of well-known films and yet Greg was entirely contrived and controlling from start to finish.
Maybe because of his teenaged-boy general idiocy, indifference, and priveleged absent-minded self-obsessing annoyance with the world, Rachel liked spending time with him and enjoyed watching the films he made with Earl because it was an absurd and outlandish take on the real world; the real world that was currently abandoning her to wallow in doom and sadness as she's being all-but-invisibly demolished by cancer.
One of the many, many montages in the movie shows Greg looking back on all his time with Rachel and reimagining all the moments where he could have -- and maybe should have -- "made a move", to bring their friendship to the next romantic "level", but he didn't, she didn't, and he didn't really know if it was ever even on the table. Greg had, almost unbelievably, absolutely no idea what was going on, and in the midst of all of that, he had this sea of emotional confusion that he never really understood and never even attempted to express. Perhaps that, too, is something that Rachel enjoyed about spending time with him, because she could see him, see into him, and watch him "struggling" to experience life and grow and be good. If the movie depicts anything concrete about Greg, it's that he has insight and artistic vision -- a subjective quality of external worth, to say the least, but he was definitely portrayed as having it in spades.
Rachel sees Greg's insight into the beauty and art of the world around him, but also sees how unaware and downright dumb he is about most things. She liked his work because she found it funny and silly and stupid but also kind of nonchalantly poignant.
As others mentioned, she clearly enjoyed collecting all the fluffy and kooky and weird pillows, not only because they are soft and cuddly and comfortable, but fun and silly. When life is destructively complex and depressing and overly stressful and terrifying, it can be wildly healthy and helpful to surround yourself with reminders of the silly absurdities of life and especially of people. As she's being drowned in such seriousness, Greg and Earl's films bring her joy and fun and escapist absurdity.
For the final film, to top off everything with (perhaps cloying) poignancy, Greg actually almost almost finally gets it, and instead of making some kind of slideshow/documentary "History of Rachel" that she clearly would have felt nothing about, he finally made a stream-of-consciousness effort to express himself emotionally. He crafted stop-motion scenes of shape and color, widely used throughout cinema as semantic corollaries to emotions, and expressed this kind of universal, explosive transcendence. It was all very soft and fluffy and cute but also very trippy and DMT inspired and somewhat reminiscent of Gaspar Noé and Michel Gondry, attempting to idolize and immortalize the transcendence of the single moment that is death, or "passing on".
So it's certainly not unrelated, but it is definitely emotionally driven and quite "random", because it wasn't really meant to be a narrative, but an experience.
Which is all very much meant as a flamboyantly factual summary of the intentions of that final film, and not at all an aesthetic endorsement of the self-righteous over-privileged under-intellectualized cloying manipulatively absurdly narcissistic idiocy that is the general message of anything and everything that goes on throughout the film where the "misunderstood" cavier-eating lazy idiot literally uses people, including a this young lady's tragic death, to justify his life, his "work", and the assumed necessity that he must be accepted into film school, though he clearly has absolutely no intellect, no social skills, no motivations, no compassion, and no common-sense in the least.
Abysmal character, but, unrelated to the narrative terribleness of the self-obsessed lead, that was the point of the final movie Greg made for Rachel.