Play Solo was a feature in which a player could access a place and edit it in a singeplayer mode, unable to play with any other people in the mode. It was removed some time in 2014 or 2015. When this mode existed, owners of games could always play in this mode in their own games, however, the play solo feature would not be available to the public unless said place was uncopylocked (see here.) It was similar to Build Mode. You could save your work as a model if you liked it.
The logo continued to evolve over time when the mode still existed. The most iconic logo features a green rounded rectangle button with dark green letters featuring "Play Solo" with a user looking similar to the classic-look of Builderman but with a different hat on [See logo above]. The last logo was just light green letters saying "Play Solo". Another term for Play Solo mode is "Offline Mode" as noted when you moused over the very last Play Solo button.