I just got my first car, and Im a little broke, so I did some Analysis on Fuel prices, here are the results of 86 Stations at like 550 different Times, equally spaced on 1 and a half month. I only took Data from where I live(which is Aachen, Germany) and only for Diesel, because this is mostly for me, but there should be no Reason why the Results differ for other Places and Fueltypes. Enough, lets get to the Point:

Lets start easy, by just taking every bit of data I have, sort it by the Day of the Week and take the Average. I hope you notice that the Y-achsis doesn`t start at zero, so Fuel doesnt cost half as much on a Tuesday, as it does on a Saturday.  What we see here, is obviously that buying gas on a Tuesday is optimal with a price of 1,145€ and worst on Sundays with a price of almost 1.17€, but also that the total difference is 1.24€ for 50l around an Average of 1,156€ with an standart derivation of 38ct for 50l. I think weekends are worst, because many people dont have to go to work, but I guess thats common knowlegde, so people try not to tank till monday, but, what i think is kinda faszinating, they could save way more by just waiting one more day

Ok, lets be honest, the last Diagramm was kinda useless, it can save you 1ct pro liter, this is way much better, I basically did the same Analysis i did for the last one, but this time I seperated by time and not by Day of the Week...this time the optimal time is around 22:00, which is a little bit strange, because you would think, that at this time, people at gas stations get annoyed, like they probably do around 2 or 4 a`clock. I think its because they want to lose some fuel, in case the price drops at the next morning, where the price becomes maximal around 4 a`clock with more than 11cts per liter more than 6 hours before(1.215€ compared to 1.101€). That means you can save 5.76€ for 50liter, which i guess is pretty good(but i guess you dont go to fuel station in the middle of the nigth that often). Here we have an Average of 1,159(it differs from the other Average because I dont have exactly as many Datapoints for each time/day) with a standart Derivation of 1.95€ per 50l, which means, on average you could save 2.9€ per 50l

One may assume, that the Optimal Time to buy fuel is Tuesday at 22:00, but just to be sure...here is every combination, and indeed, the best time is Tuesday at 22:00 with 1,09€, worst is Monday 4am with 1,22€ and the Average is 1,158€, so on average you could save 3,38€ per 50l

So lets recap... If you want to save Money while buying gasoline, you should:

1)Do it in the Middle of the Week

2)Do it close before Midnigth

than you can save almost 7ct per liter, which is way more than the money you save be driving to a cheaper station