Symbolism, Imagery, Allegory
Yep, our fuzzy little friend is a symbol for Charlie's status as science experiment. Let's face it: he lives in a cage and runs through mazes for food. And most importantly, the little guy couldn't exactly give his permission for a mind-altering surgery (unless he was really good at communicating through squeaks). As a helpless animal, he's totally at the mercy of the mad scientists. And he's not the only one: Charlie, too, has barely any ability to tell Strauss and Nemur no, let alone give his consent for all the things they do to him in the name of science.
"I think Ill be frends with Algernon" (8.32), Charlie says at the beginning of the book, setting us up for a whole lot of comparisons between mouse and man.
Sure both of their lives have been forever altered by being part of the same experiment, but the fact that Charlie is always being compared to Algernon shows how they both are viewed as animals first, and living and feeling beings second. Charlie tells us his every waking thought, so it's hard to think of him as animal, but Strauss and Nemur sure like to remind him at every chance that he was nothing before they operated on him. Charlie looks at Algernon and sees, well, himself.
Even after Algernon's death, his body has special significance to Charlie, and it's not like he's just going to let Nemur stick him in the incinerator and call it a day. Nope, Algernon's body is the common denominator for all the versions of Algernon: ordinary Algernon, super-smart Algernon, and losing-his-grip-on-reality Algernon. While Charlie is desperately trying to hang onto his own life, keeping Algernon's body in his possession makes him feel more in control. If he can't stop the inevitable loss of intelligence, at least he can put some flowers on his best buddy's grave.
Even more importantly, Charlie wants other people to commemorate Algernon and what he meant to the world. We can already hear you shouting it out: Charlie doesn't want to be forgotten, and he views Algernon as a little piece of himself. Putting flowers on Algernon's grave doesn't have to mean literally picking some dandelions from your mom's garden—even just saying his name, for Charlie, helps keep him in the world.