Standard garden tap size (US)

Oil's no good, water will wash it away, use silicone grease. You may be able to grease it up without taking it apart. Turn the water off, then lift the handle all the way up. Put a bit of grease on housing exposed when the handle is lifted, use a cotton swab to get into the gaps. Exercise the handle up and down and see if that gets rid of the squeak. If that doesn't work you are going to have to take it apart (look for an explosion diagram of your faucet online), take lots of photos and keep the parts in plastic containers to make sure you don't lose anything. Check the condition of the washers as worn washers may cause binding, hence a squeak. When putting it back together grease the parts that rub against one another as you go. Don't go overboard on the grease, a little goes a long way.

One word of caution: before taking anything apart it's important to realize that you may not be able to get it together again, and may require that you replace the faucet entirely. Some units are not meant to be maintained, may have plastic pieces that break and you can't get them back together, or may have proprietary washers or other parts that you can't buy replacements for. If it's broken already then you lose nothing by trying of course.

It is also a bad design because any water that gets in there pools. For that reason I would "glue" it on with a basic latex caulk. When you need to service the tap you'll have to cut out the caulk.