I'm wondering if Lightspeed Flash has the power to travel at the speed of light, or is faster than it, or is it just some kind of hero name?
Flashy Flash hasn't shown any of his abilities or powers in the manga or at the tv show. But in the webcomic series of ONE (the original manga), which it has a lot more chapters, we know some of his abilities and his powers.
So according to the webcomic:
One of his moves, was to fly to the enemy within milliseconds and attack him from any distance. Flashy Fist (閃光挙, Senkōken): In only a matter of milliseconds, Flash traverses the distance between himself and the opponent and unleashes a barrage of punches.
So you have to wait until to the point where the manga, will show more about this character.
In short, it's just a hero name which is flashier than reality. Pun intended. Some more recent translations have switched to using "Flashy Flash" instead of "Lightspeed Flash" specifically in order to avoid the light speed comparisons, and to better reflect the silliness in his Japanese name (Flashy Flash is a fairly literal translation).
In a couple pages of the webcomic we're actually given a timer on one of his actions, which allows some rough estimates of travel and attack speeds. This forum post (which is itself a summary of a post somewhere else but which I don't have a link for) contains the calculations on said pages, and estimates both his movement and attack speeds (arm speed during a punch) at around 2.3% of the speed of light, which isn't even fast enough to warrant relativistic corrections. It's hard to say what the error margin is on those, for a variety of reasons, though it should be well within an order of magnitude—20% of the speed of light would warrant the use of special relativity, at least. However, the calculations also assume that ONE's depiction of the scene is reasonably close to scale (and his drawing skills make that questionable), and that he actually drew it to that scale after deciding how fast he really wanted Flashy Flash, aka Lightspeed Flash, to go. The scene in question also occurs during a very serious sequence where Flash knows that many of the best combatants in S class have been defeated, so it is reasonable to assume this is near or at his maximum speed.
There hasn't been an official description of how fast he truly is, so this is the best we've currently got on his capabilities.
Here are the webcomic pages in question: