The structure was built in 1959 as per Brazos CAD, built as a house, the first reference coming from a 1963 obituary about one Mrs. Marguerite Mary Edmonds, age 48, and at some point in the 1970s it was abandoned.
Dead Solid Perfect was a hamburger restaurant and the first that opened in the former house in February 1976. According to this article, [owner Searcy Bond] did nothing to the building" during conversion to a restaurant, which matches up with this description is from Yuccadoo on TexAgs, with "no exhaust vents, smoke pouring out a coupla windows in the kitchen". It closed in 1979.
In December 1979, One Potato Two Potato opened; I'm guessing it was the first "meal in a peel" style restaurant in town. In the summer 1983 it was retooled as Two Blocks North and added hamburgers to the menu. In 1983 Two Blocks North closed, and in August 1984, La Taqueria & Tortilla Factory opened. "La Taq" was arguably the most memorable of the restaurants in the space and is still talked about sometimes today. La Taqueria closed around summer 1991. According to Aggieland 1992, Rosalie's Pasta House opened September 25, 1991 in the former La Taqueria space (and had the same ownership). Around 1997 it closed.
Finally, La Bodega Baja Taco Bar opened in October 2000 (likely with a major renovation that brought it up to code), and closed in November 2014. I do have a picture of the specials as of May 2014 including the gimmick of the hotter it was outside, the cheaper drinks were.
Despite some talk of more upscale dining in the Northgate area, it and the adjacent apartment building next to it (104 Church Avenue, where Eccell Group operated out of) was torn down. What has replaced it is supposed to be a food truck park, called Wayside Food Park. It was supposed to be a permanent place for food trucks complete with electrical conduits and a covered pavilion area, but it failed hard, opening in December 2017 and closing by early 2019. In regards to that apartment building, I have terrible pictures of that, the one with my finger covering a quarter of the shot is from this May (2014) and I thought had a better picture that shows the building as a whole but I can't find it. It's not much better than what you can see from the older shots on Google Maps Street View.
UPDATE 04-26-2024: Article rewritten for better prose and better information. This article was previously amended in May 2019 to account for the coming and going of Wayside, but now it and the adjacent 100 Church Avenue have been knocked down for a new student housing complex. On April 22nd 2024, the article was been renamed to "A Look at Former Restaurants at 102 Church Avenue", with [demolished] also added.
