Simply go to the send tab, and in the "Pay to" field you enter OP_RETURN followed by your message in hex. So for your string "Hello, world!", your entry would look like this:
OP_RETURN 48656C6C6F2C20776F726C6421
Then just set the "Amount" to 0 and hit pay. You'll get the usual Electrum pop up to choose your fee, etc.