Few things that people can check by just looking at their own blood under the microscope, if they take some time to learn the skill. Counting red and white blood cells, for example. But without a lot of practice, those numbers could be very wrong. Also, you probably won't be able to differentiate between the types of white blood cells.
There are also some things that can be done using tests that you can buy at a pharmacy (and maybe even a supermarket), for example blood sugar. These are usually very easy to use so people can use them at home. When in doubt, ask a pharmacist or doctor.
Some bacteria can be stained and identified with equipment that can be purchased from specialised businesses. Again, this takes practice. It's easy to get something wrong.
And unless we are talking about diabetes or leukemia, both of which would also show up when being tested at a doctor's office, these tests are not helpful in your scenario. Almost all other medical tests, like testing antibodies, hormones, vitamins, CRP, need specialized lab equipment and people with training.
There would be the possibility of taking blood and having it analyzed later. But while blood can be stored in a cold environment and analyzed later (most doctor's offices I know only deliver to a lab once a day, storing the blood in a fridge before that), most people aren't trained at taking whole vials of blood, and even people who are don't usually take their own blood.
The patient's best bet in this scenario is to have some doctor take the blood while the symptoms are occurring and send the sample to the lab. Doesn't necessarily have to be the treating specialist - one of mine is 70 kilometers away, so I have my blood tests done by my GP I can walk to.
As a bit of an aside, I have a hard time coming up with anything that would show up in the blood sample while showing symptoms, but fail to show up in one taken a few hours later after symptoms subside. Then again, I am not a doctor.