Here’s the breadboard components converted to a small 45.5mm x 35.5mm PCB board design using the free version of Autodesk Eagle. The files will download via the green button below and are also on GitHub.
Two female 2.54mm headers (12 & 16 pin) accommodate the Feather. Two male 2.0mm JST PH shrouded connectors (2 pin) receive the LED and solenoid. I etched the board myself using a laser printer and muriatic acid.
I've posted my etching instructions. If you don't fell like doing it yourself there are many online services that can prototype a PCB for a very reasonable fee.
You could also skip the design and just use an Adafruit perma-proto board.
Female header pins, cut to the right length with diagonal cutters, make for a socket to plug in the Feather board for possible changes.
For the rest of the circuit, move the parts and wires from the breadboard to the corresponding places on the Perma Proto and solder on, it's that simple!