So anyone have suggestions or even another system they have come up with or know of to suggest?
I would generally advise against a "system". Even if you plan to have a riddle every session, you'll be better off handling each riddle in its own way rather than trying to make it fit with a chosen system.
Then I think your idea of granting hints towards the solution is of course fine... it's quite normal to use for example Knowledge skills for hints about the story/plot or anything else, so why not also for solving a riddle where the players are stuck. How many checks/hints? How high DC? Which skills apply? All that stuff better be decided for each riddle and the situation around it. Just setup the practice for yourself when you prepare the gaming session.
The only bit I would probably settle for in general, is to use group skill checks instead of individuals, because this is very likely always to be a case where success or failure applies to the group as a whole.
I think it would be much better in general, if you deploy a riddle with a cost for failure around it, when using the hints checks. One idea could be to link the riddle to a trap, with each failure triggering a chance for the trap to spring: see the famous organ-riddle-trap scene in the movie "The Goonies". This could be such that each time the players give a wrong answer and/or ask for a hint check they trigger such chance. Another idea could be diminishing returns, so that the prize for solving the riddle decreases.
The main challenge is tying these with the narrative. If you have a Sphinx delivering the riddle, you could easily have it explicitly tell the characters that she will give them less of what they need for every failure. But of course this requires that what they need is quantitative, such as a magic item or information, if the prize is simply granted passage then it would be harder to diminish it.
What if instead the puzzle is not delivered as a question by a creature? How are the characters supposed to deliver their answer? My favourite type of riddles are the mathematical ones, and in that case I try to deploy the riddle in the form of a mechanical device that needs to be setup properly to represent the answer. If the answer is a number, my typical solution is having a plate or container with a counterweight mechanism, and a number of pebbles/coins/etc available, and the characters must put the correct number of them on the plate and pull a lever. You can then have all sorts of bad stuff happen when the number is incorrect. If the answer is a word or a specific order/permutation of items, you can have small objects representing them (e.g. small figurines of things the name initial of which can be used to "write" the answer).
All that doesn't necessarily have to do much with the "hints", which you can allow independently from how many tries at the answer are allowed. You can in fact just manage hints as the characters thinking over and pondering the answer before even trying it. In that case, time can be the cost for asking each additional hint, but this works only if time is an issue at all in the current adventure, otherwise it is not really a cost. The only risk to watch out for here, is trying to avoid the case when the players just ask for all the hints possible because there is neither a cost for that, nor an advantage for solving the riddle without hints.