I have several things that I would like to see in a new Rayman game:
1. A better continuity. It can be very jarring to see established elements and concepts of Rayman's world change so drastically between each game. I'm not trying to stifle the creativity or anything, I just want a little more consistency. It also tends to bring about a lot of unanswered questions (Where is the Valley located? Where are the islands from Origins located? What's the deal with Tarayzan and the Musician? Why did the Magician change species? What's with the inconsistencies regarding the Knowledge of the World? What happened to the Great Protoon/Is the Protoon the same thing as the Heart of the World?, etc).
2. A 3-dimensional set-up. I don't mind 2-D, but I feel that there's a lot you can do with Rayman in that set-up. It's a great mixture of open and linear gameplay that can give a sense of adventure, magic, suspense, and omen (more-so in Rayman 2 than 3, though).
3. A return of characters such as Ly, Clark, Eig, Sssssssam, Joe, and others. Since Betilla has returned as a major character, I would love to see a rivalry between her and Ly. Maybe Ly could be from a different group of fairies that don't get along with nymphs or something. Also, Betilla should act as a "mother" for Rayman while Ly should be a good friend. Clark would be a nice addition to give the series more of a main cast than recent installments. Ssssssam and Eig could be either friends or enemies and act as the main characters of the swamp levels. Joe could tie into a concept that I will introduce in the next section.
4. The alien race from Tonic Trouble playing a role in the plot with The General as either the main villain or just a major villain. Yes, Tonic Trouble sucked, but the concept of these limbless aliens could be very useful in the series. One reason being that they are, well, limbless. This could add a concept of there being more than one limbless race and more than one ways for limblessnes to occur. Another reason is that The General was supposed to be the villain of Rayman 2 back when it was being developed as a 2-D game, and has been established in Rayman's universe (which, in turn, established Tonic Trouble within Rayman's universe, at least in my eyes). Also, I would just like to see Rayman beat the crap out of Ed. Heck, maybe Ed could defect to Rayman's side at some point and become a regular character or something (or maybe not). It definitely seems like Ubisoft remembers Tonic Trouble, since Gourmand Land in Origins has elements of that game in it (the martini/cocktail theme and the fire-breathing chili peppers to name a few). I don't know, I think it could work. Also, Joe could set the whole alien thing in motion (maybe he could have connections with the race or something).
5. The muse/ex-wife of Polokus playing a role in the plot. Rayman might have to search for her in the forest to get her aid and maybe help her see that the Teensies aren't as bad as she thinks.
6. The People of Prhys. This could help give a back story to Ly and explain why she is the way she is.
7. The Knarren and Leptys. I would love to see a back story for Leptys and the Knarren as this could open a whole new element to the series: gods. Now, Polokus is considered somewhat of a god already, but it is mentioned (and shown) that there is more than one god in Rayman's world. Establishing these gods and giving little bits of information about who they actually are can give this world more depth while still retaining a sense of mystery and wonder. Also, the Knarren are a good addition in terms of threatening races, so it would be neat to see them again.
8. A relatively more serious tone/plot-line (without sacrificing the humor, of course).
For a completely unrelated game, how about the "first nightmare" as a basis?
Anyway, that's what I would like to see in a new game. There's probably more things, but I can't think of them right now.