From a more symbolic perspective, there's a rather large group that insists Chu-Chu is the embodiment of Anthy Himemiya's personality.

This does not seem reasonable unless you've seen the entire series and realize that beyond her carefree façade Anthy is in constant pain and despair. The general theory seems to be that Chu-Chu is what Anthy would be like if she could break away from her role as the Rose Bride. And in fact, the entire series is essentially about Utena's quest to free Anthy from this position.

A good example often used is episode 27: Nanami's Egg. Nanami and Anthy are polar opposites from the beginning; both play younger sisters with questionable big brother relationships; yellow and purple, their respective colors, are on opposite sides of the color wheel. (Color symbolism, by the way, runs rampant in Utena and most other anime.) While the episode appears to be a filler, a basic rule about this series is nothing is done without a reason.

So it is inferred that Chu-Chu hatches out of Nanami's egg, or is reborn. This parallels Anthy's growing fatigue which probably relates to her performance throughout the black rose saga, which I won't ruin for those who have not seen it in its entirety.

A few notes about Chu-Chu's character design: Chiho Saito's manga features a few changes, mostly he's fat and much more mouse-like. Chu-Chu itself is a sound effect that mice make in the Japanese language; there is also a character called Chu-Chu in the playstation game Xenogears. The big giveaway to me as to his real identity has always been his big naked rear end, which was definitely added in order to facilitate his status as a monkey.