Interesting question. I am gonna go with it really starting during the second term that Grover Cleveland served as President - between 1893-1897 - as being the point when the first White House daily press encounters between White House "clerks" (I don't think they were called "press secretaries" in those days) and the press began to be a normal and expected occurrence.

This idea of the press having daily access to official information from the White House seemed to have largely been kicked off by a guy from South Carolina who wrote for the "Washington Evening Star" named William Price, who began calling himself a "White House Correspondent" back in 1895. By my understanding, this reporter dude (Price) began stationing himself at the White House every day, constantly asking questions of the White House clerks (guys like George Cortelyou - then stenographer and executive clerk at the White House under the Cleveland Administration). Once Price began to get the "scoop" on several stories, other publications rapidly took note of his access, and began sending their own reporters to the White House to dig for information/stories as well, making White House interaction with the press a pretty much daily occurrence - both officially and even unofficially. You could go back earlier than that, but it seems a lot of communication from the White House to the press before then was much more sporadic.

Officially though, I think it was under the Theodore Roosevelt presidency that the more concrete idea of a dedicated White House press liaison was implemented. The guy who was tasked by doing this was the same guy Cleveland had utilized as his official stenographer and as an executive clerk - George Cortelyou. Cortelyou also worked for President Mckinley after the Cleveland Administration, then stayed on under Roosevelt. He is recognized as the guy who set up the procedures for the office of the presidency to communicate with the press directly - first under McKinley's term and then under the Teddy Roosevelt Administration, and was the guy who largely handled the briefing of the press (first somewhat unofficially and then officially) - even allowing the press to set up their own base of operations (a "White House Press Corps", if you will) within the White House itself.

So, George Cortelyou is the guy who seems to be the one who got the ball rolling on the concept of official press briefings coming from the White House on a regular basis to keep the public informed, really becoming official briefings during Teddy Roosevelt's terms. Whether these briefings were literally "daily" or not at that time I do not know for sure, but Cortelyou and his staff were known to have had very regular interactions with the press while working for Cleveland, McKinley, and Roosevelt, but it was officially under the Teddy Roosevelt Administration (from 1901-1909) that very regular White House briefings to the press became both common and expected.