This building is unremarkable but has anchored the Northgate for years (Street View here). The lot was vacant in 1961 with no parking (possibly a former storefront that was burned/torn down years earlier). The modern building was built in 1968.
The first reference to the building I can find is an establishment known as Iva Starnes Men's Wear, which had locations in Bryan (and Houston, at "Fondren at Westheimer", though no exact address) in 1969. In 1971, it became an establishment known as "The Pub" (though directories refer it as "The Tavern"). In late 1973, Kesami Sandwich Shoppe ("Where No Two Sandwiches Are Alike!") replaced The Pub, which lasted until around 1976 and by summer 1977 it was Farmer's Market Sandwich Shop, a limited menu version (it later became full line) of Farmer's Market Bakery and Delicatessen in Bryan. It sold sandwiches, soups, salads, and fresh baked goods before becoming Gideon's Farmer's Market (adding catered meats to the Bryan location and pasta, pastries, and beer to the Northgate location--free delivery to dorms, too), and then closing the Northgate location, which became Equinox (the Bryan location closed later). It was somewhere in the long history of being Gideon's/Farmer's Market that it absorbed 331 University (likely 331 University used one of the dual entrances in the building, though Basil Whippet's removed one of the entrances).
The upstairs area, 331 University, was home to smaller offices and tenants. Shamrock Employment Service (Shamrock Employers) was here in 1969, and by 1972 (according to a city directory, there were three tenants, including a vacancy, Guinn & Co. (insurance brokers) and Kentucky Life Insurance Co.). Other tenants over the years included Homer B. Adams Realty Co. (1972), The Shape of Things (1975-1978, hair salon), Bear Paw Turquoise & Silver Jewelry (1975, jewelry). In 1984, it was the home of On the Double copying service.
At some point, 331 University was absorbed into the main building.
By 1993 the lower level was Equinox, and by 1999 it was Crooked Path Ale House. Somewhere between this time it was also Northgate Café, as well. In 2005 it was The Library. While having a bar with a name like that isn't too unique (seems like Michigan Tech had the same idea), The Library closed in 2008 or 2009 due to poor sales. BMO's, a rap-oriented bar, opened in the spot in 2009. It ultimately did not last more than a year. In 2010, it opened as Basil Whippet's Pub & Apothecary Lounge, and after closing in 2016, became "Icon Night Club & Lounge". The building has had a few exterior updates over time as well.
UPDATE 01-05-2020: Updated post to account for former tenants, including original stores inside.