The purpose of using a roux, as opposed to just plain flour, is to improve the dispersal of starch molecules in the sauce. If you just toss a bunch of flour into a simmering sauce - or do the reverse, pour hot liquid onto dry flour - then you'll immediately start getting gross glutinous lumps and will find it nearly impossible to smooth them out.
Starch is not actually soluble in fat; instead the fat coats the starch molecules, making it difficult for them to clump together when incorporated into a sauce. Making a roux out of the flour (and making sure to give the sauce a really good whisking after incorporation) basically helps to ensure an even distribution of starch so that your sauce ends up smooth, not lumpy.
So to answer your question, the minimum amount of butter (or other fat) you need is however much it takes to fully incorporate the flour. If you've still got dry clumps, you haven't used enough fat. More fat is okay, but if you don't use enough then you'll suffer some of the same lumpiness as if you didn't use a roux at all.
I'm not sure if there's a magic number - it's going to depend on the specific flour you're using and the specific fat. Even if you narrow it down to, say, all-purpose flour and butter, it can depend on the brand of flour, the fat content of the butter, and whether or not the butter has been clarified (which, if you have the time and patience, it should be).
The 1:1 ratio (by volume or mass) gives you a pretty decent safety margin; you won't need more than that, but it's also not so much that it'll make your sauce greasy. But you really don't need to measure it out at all; I don't, I just add fat while stirring until the flour is fully incorporated. Sometimes I might be more generous with the butter simply for flavour purposes.
Bottom line, it's really pretty obvious when you haven't got enough fat. If it looks good, it is good, and after a few sessions without the measuring spoons, you should very quickly get a feel for how much fat you really need.
If your sauce is too thin, the problem is that your initial roux was either too thin (not enough flour) or you added too much liquid for the amount of roux that you made. Standard ratios are 1 Tbsp butter - 1 Tbsp flour - 1 cup liquid for a thin sauce, 2-2-1 for something in the middle and 3-3-1 for a thick sauce.
Once you've made the sauce and it's too thin, the best fix is to add a beurre manié, which is essentially butter and flour that you mash together in a small bowl and then whisk into your sauce. It's kind of like adding more roux after the fact. However, because it's not pre-cooked like the roux, you need to make sure you give it enough cooking time after adding to get rid of the floury taste.
Best Answer
Cook it longer, and watch your proportions. If it's too dry (not enough fat), it's hard to cook through without burning it. You want to get to a golden brown color throughout. You can cook it until it's darker and it'll add more caramel/nutty flavor (don't burn it), but it has to be at least a golden color before it's cooked enough to not taste of raw flour. When in doubt, taste it. If it tastes raw still, go a bit longer.
A good start for proportions is 1 part flour to one part fat--by weight, not by volume. Start with your fat in the pot, and get it nice and warm but not blazing hot, then add your flour a bit at a time, stirring (or better yet, whisking) for a couple of seconds to incorporate it smoothly before adding more. Stop adding (no matter how much flour you've used) when you get a slightly soft paste. Not too runny, but not as stiff as toothpaste either. I've yet to find a common thing people can agree on that has the right consistency to describe it, unfortunately. When you get the right consistency you need to keep stirring it over the heat until you get that golden brown color.
I admit that it's a little imprecise, but it really is a thing I do by feel and experience, based on what I said above as my original starting point. I've learned how I like it over the years, but I started with that.