Burnett park serves as a nice getaway for office workers of Burnett Plaza and 500 W. 7th, and a nice place for residents of the Neil P and Electric buildings to relax. I think it could be a lot better, there's not much going on with it. If one wanted to enjoy lunch or something in the park, they'd have to get it elsewhere and go into the park.
My simple idea is to eliminate the right-hand turn lane from 7th onto Lamar and the sidewalk island it creates and replace the entire section with a small standalone cafe.
This location would be highly visible from vehicles on the busy 7th street, as it sticks out a bit (perhaps some bollards would need to be erected to prevent any crashes into it) and would be easy for pedestrians to reach. The location is within 2 blocks of several large office buildings, (Burnett Plaza, Fed Courthouse, 500 W 7th, Federal building, ST building, FW Club building, Frost Tower) and immediately across from around 150 residential units.
For the architecture of the structure itself, I imagine something with an open and glassy feel, with lots of outdoor seating in the park. Good examples of in park cafes can be found in Houston's Discovery Green or even in Klyde Warren park.
Ideally, the tenant would be a local brand and serve breakfast & lunch. I don’t believe this area has enough pedestrian activity after 5 to support a dinner service. Perhaps there could even be two tenants, for example, an arrangement with a primary food service place and a smaller spot serving ice cream or other desserts. In a lot of ways, I think the Woodshed would be a good example, just in a more parklike setting and without all the tax issues.
Overall. I think such an amenity would liven up the park and make much better use of the existing conditions.