I have my respect for Mr Tesla, but it seems weird that "he" was chosen to be the units of magnetic flux density. I mean, he didn't contribute much to magnetic fields theory, nor did he work with them a lot. So why was it named after him? Why not, for example, after Hendrik Lorentz, who formulated Lorentz Force?
Actually, Tesla's first major invention was based on his discovery of rotating magnetic field around 1882. To quote colorful description from 1919 Electrical Experimenter:"There is a vast difference between an ordinary electro-magnet and that invented by Tesla. In the former the lines are stationary, in the latter they are made to whirl around at a furious rate. The first attracts a piece of iron and holds it fast; the second causes it to spin in any direction and with any speed desired." Look at Tesla's Egg of Columbus video to see what it can do. This was the source of all modern AC motors, Tesla patented the first one in 1887, that before him were believed to be impossible because "the alternating current was but imperfectly understood and had no standing with engineers or electricians". Nuclear magnetic resonance underlying the modern Magnetic Resonance Imaging (MRI) technology is based on properties of magnetic nuclear spins, a quantum analog of rotating magnetic field discovered by Rabi in 1937. Modern MRI machines are calibrated in tesla units.
A word on terminology, technically tesla is not a unit of magnetic flux, that would be weber, but a unit of magnetic flux density per area. The genesis of tesla was somewhat controversial, disagreements over whether magnetic field strength and magnetic flux density were quantities of different nature lasted until 1930 when the International Electrotechnical Commission declared that they were, and tasked the Technical Committee 1 to come up with new units. In 1935 it passed the task to the Technical Committee 24, which dragged its foot until 1956 (in fairness World War II happened, and they had to settle on units for several other quantities as well).
There is a unit named after Lorentz, but a very obscure one, "a unit of reciprocal length used to measure the difference, in wave numbers, between a (zero field) spectrum line and its Zeeman components". It is fitting since he is the founder of the (classical) electron theory. But Lorentz is honored much more meaningfully by having the electromagnetic force acting on charges named after him.
According to this article, the name 'tesla' was first suggested to the Technical Committee of the International Electrotechnical Commission in 1950 by two professors from the Belgrade Faculty of Electrical Engineering. The suggestion was accepted by the committee in 1954, and by the International Committee for Weights and Measures in 1956, which also happened to be the centenary of Tesla's birth.
The choice of 'tesla' was publicly justified due to Tesla's "fundamental scientific work" in the "electrical world", rather than anything more specifically connected to magnetic flux density.