As the title says, why did Eobard Thawne (AKA Reverse Flash) will his properties to Barry Allen, despite Allen being the arch nemesis of Reverse Flash...?
I don't believe that it was ever outright stated in the show, but these are the two reasons I saw:
It was his chance to get one last dig in at The Flash
Part of the condition of the will was that Barry was to watch his final video. This let him deliver a final monologue, and get the figurative last word in on his opponent, certain that Barry will never be able to deliver a counterpoint to him.
He seemed to become genuinely proud of Barry before his death
While it was not enough to deter him from his plan, he seemed to take genuine pride in Barry's accomplishments, and to view him as the son that he never had. If nothing else, he saw Barry as one of his greatest accomplishments, and both genuinely wanted to give him one final gift in the form of the confession, not to mention giving Barry the resources to continue to grow.