Once in a while my dad would remove the needle valve and spray it down with carb cleaner. He never used Stabil, but the last time he used the mower in the fall he would let it sit and idle until it ran out of fuel, then put it away for the winter. - this way there was no old gas left in it to create varnish over the winter.
If the mower has been sitting for several months with gas in it it is very possible that the gas has gummed up the carb, but the carb cleaner should take care of that if that's the problem. Stabilizer is a good idea if you're going to leave gas in it between seasons. I always fill my motorcycle up with gas in the fall and put stabilizer in it then run it a few miles to get it all throughout the fuel system. Usually that tank will last for what little riding I do during the winter months, but if I need to refill I just add more stabilizer.
P.S. Feel free to laugh at me for being completely ignorant of this stuff...
Ok, now that that's out of the way, someone else mostly answered your question. To rephrase and add a little...
- If it's a 4-stroke engine (most likely) you don't mix oil into the gas. Look for a tag with the model and serial number to look it up and find out if it's 2 or 4 stroke. Look for an oil plug/fill hole, maybe near the bottom of the engine; if there is one then it's probably a 4-stroke engine. Change the oil, it's probably awful by now.
- If it's a 2-stroke engine (possible), you probably will have to mix oil into the gas. You must use oil made specifically for the purpose. You'll have to find out what ratio to mix it at; that same model number you looked for as I suggest above could help, or it may be marked on the gas tank, or it may be a mystery. The nice thing is that if you know the ratio, you may be able to buy 2-stroke oil that's prepackaged in measured bottles so you just dump a whole little bottle of it into a gallon of gas.
- If it's a 2-stroke engine that mixes the oil automatically (highly unlikely), you'll need to find out exactly what type of oil to use with it.