It's commonly asserted that ideas are worth next to nothing, the difficulty is all in the execution, etc. I don't think that's entirely true. If your idea is just a vague one-liner like "facebook for pets", then sure, it's not worth much. But as you start building out a prototype, you're forced to be more specific and refine the idea, and it becomes more valuable. If you've got a handful of users and you've gotten some feedback, that's yet a bit more value. And if you've thought about the problem for a long time, you will probably have a good sense of what are the main challenges and what approaches won't work. That's valuable too.
I don't think there is a well-defined line between refining your idea and "executing" on it. Obviously there is a limit to how much you can accomplish in the planning and prototyping stage. However, I suspect it is a lot higher than is commonly believed. Working for 3 months may sound like not very much on the scale of a startup's lifetime (let's say 2-5 years before it either fails or grows big), but consider this: maybe the OP did 4 or 5 failed 3-month projects before he or she got to this one, which seems somewhat promising. Suddenly that looks a lot more significant than 3 months.
For this specific case, we don't have enough details to assess whether the OP has really hit on something big, or if there is still a lot of uncertainty about whether the idea will pan out. So unfortunately, after all that discussion, I don't really have an answer to the original question. I think what I would do in your situation is to make my best guess as to the "fair split" based on what I've contributed and the expected future value (let's say I decide it's 70/30), offer the friend that split, and if he/she accepts, I would immediately change it to 65/35. That way, they should be happy, but things are still close to fair from my perspective.