You may find SQLite to be a good light-weight alternative to MySQL. It is a much "lighter" install. It may not be great for heavy-lifting, but it has essentially no limitations on size of data or database, and takes a simple installation of library and command-line tools on your backend system. Each SQLite database is contained in a single file. There is no "SQL server" per se. You just call the library from your server code. The database code runs in-process in your server code. (Instead of making requests to a separate SQL server.)
You didn't mention what server language you might use. A lot depends on what language(s) you know or think you might easily pick up.
node.js would let you use Javascript on your server.
My preferred solution for quick/easy/small servers would be Ruby with Sinatra and Sequel. You can write a simple service in just a few lines of code, and even with database storage you can write truly useful servers in 100 lines of code or less.
I have a simple server that accepts JSON POSTs and stores them in an SQLite database, written in Sinatra. It is in 70 lines of code!
With node.js/Express, you might use sequelize, though I think it is much more difficult to get started with it than ruby/Sequel. You have to have a good handle on Javascript Promises to use sequelize.
They are both very similar environments. Express was modeled after Sinatra.
FWIW, I like Db Browser for SQLite for "manually" working with SQLite databases. It's available for Mac/Linux/Windows. You can easily view and sort tables, add/remove data, try out SQL commands. There are a bunch of others, (as well as similar for MySQL or PostGrsel or whatever) but I like it's simplicity.