How can I find a family doctor?

You will need private insurance as even HU citizens who moved out of Hungary and didn't pay into the system for at least one full year in advance can not use the National Health.
For those HU citizens who moved out of Hungary but kept up the monthly payments into the system there would be no discontinue of coverage.
Perhaps your husbands job has coverage for the family?

I second what Marilyn said.

And, I am a little confused. How did you get an address card and thus I assume legal residency in Hungary without proof of existing health insurance for you and your family members? Proof of existing insurance usable in Hungary is required by law to get residency.

In any case, you need to either talk to your current health insurance provider about coverage for your child, or contact NAV (the national health system) to see if you or your child is eligible or not at this point in time

And you can not really "select" your doctor under national health in most cases. You go to the local clinic and get the doctor they have there that day.

I was also wondering how they got a resident card without proof of health insurance.
Then again, I don't get how things go here in Hungary.
I always see to come up short and get the hassles with anything official here while some people skip right on through.
Maybe having a job that is willing to pay for insurance is another matter?
Perhaps they are coming in from another EU country and have insurance in their homeland?
All I really know is I pay for NAV even when I am out of Hungary and can't use the coverage.

In the short term it would not be that expensive to pay as a private patient for minor ailments. The same applies for dental treatment, prices are very reasonable and a lot of Hungarians go private just to get seen quicker.

I don't know if it works for non-EU nationals or if covered by the private insurance but if a woman had the TAJ (HU national health) card and you've had a baby in the past 3 years it used to be (and probably still is) that that you and the baby would also be covered at the same time.   Mrs Fluffy says also that the kids are covered under her TAJ card until they are 18 as they are in school.  It might be the same even if through the private scheme.  Best to ask more questions.

You can select the local doctors in the village but you need a children's doctor as well so you may have no choice other than the only pediatric doctor available. In my own village  we have about 6 doctors in a few GP surgeries and as far as we know, we can see any of them.

For dental, it's always been private for me and Mrs Fluffy although some people do use the government dentists.  We also use private doctors to skip over the waiting time.

If you do have a National Health Card you must show your address card because you must first go to the medical clinic in your district and first see a family doctor.
You must show the address card because you can only visit the clinic in your district.
From there your family doctor will send you on to any specialist they think you need.
I tried taking a short cut by going directly to a surgeon, got a 2 month run around and learned it would of been more efficient and I'd actually see the right doctor for my condition if I had just gone to my trusted family doctor first and not tried to take a short cut.
There are no short cuts with national health.
One tip however is to stay in the good graces of your family doctor and show them how valuable they are by giving them a "toke" tip, token of appreciation ...
Sometimes they will refuse but you should offer, most people we know tip between 5,000 and 10,000 for a visit, in the long run it could be almost the same cost to just see a private doctor and not tip them.
A specialist  should be tipped more, I tipped 20,000 to get my shoulder examined and have the ortho doc drain my shoulder with a needle, took less then 5 mins of her time but was worth it to me after 2 months of running around getting nowhere with the one surgery clinic.
We wondered why they were giving me a hard time with my obvious issue with my shoulder at that one clinic in our neighborhood.
It was because they must keep records on who tips and how much because years before I went in for a super minor thing and didn't tip the doctor. Not that I wasn't going to tip him but he made a minor issue into a major event and I had no idea how much to give him for his side show , He booked a operating theater and had a nurse help him all for a tiny little blood blister on the top of my foot, he was literally huffing and puffing like he was dying because he was working so hard on my foot.(dropped something hard on my foot and it let a tiny mark which I wanted removed.)Got a bad rap for not tipping him at that surgery clinic.Of course they will never admit that was the reason they sent me here and there with no results for over 2 months with my shoulder but we aren't that naive, money talks in Hungary even for those who one would think are above tipping.
When I went into the surgery with my shoulder they were all friendly and good to go until they looked me up on the computer... Bad tipper must of been in large letters because straight away their attitude changed and all of a sudden they couldn't help me because they were going on holiday, come back next week and see another doc.
Neighborhood clinics come in all sizes depending on how large your district is, they have several family doctors to chose from. However once you pick a doctor you must stick with them for a full year before changing  your primary doctor if you aren't happy with them. Think it may be a written request to change but not sure about the details on changing a doctor.
We notice that often many locals mostly young families and the elderly look like perhaps they can't afford to tip much or at all. My husband thinks the doctors let the tipping slide with these people but as a ex-pat you had better come up with some gravy or you won't get the care you should get. Just the facts of life in Hungary as an ex-pat, they can tell if you're from the west and expect to be taken care of for their services. It's a touchy subject and most locals will not even tell you how much they tip or talk about tipping. It's like a dirty little secret,

Marilyn Tassy wrote:

.....
When I went into the surgery with my shoulder they were all friendly and good to go until they looked me up on the computer... Bad tipper must of been in large letters because straight away their attitude changed and all of a sudden they couldn't help me because they were going on holiday, come back next week and see another doc.... It's a touchy subject and most locals will not even tell you how much they tip or talk about tipping. It's like a dirty little secret,

I expect they've got some doctor shorthand in there as illustrated here:

Doctors' Shorthand



BTW, HUF 10K seems extraordinary expensive for a bit of early poking about.  If they were gassin' and slashin' (according to the article anesthesia'ing and surgeon'ing) then may HUF 20K would seem cheap for paying a bit of attention.  Not really very good that tipping is even remotely on the radar.  BTW, we don't tip for kids so yes, I think there's a hidden etiquette there for the have and have nots.

We usually tip our family doc 5,000 for a visit,I go along about 50% of the time otherwise my husband has his own appointments.
I only go because I figure if he has to go in for a blood test I might as well get a bit of bang for the buck even though I am in good health expect for my joints where is both wear and tear and probably all related to the immune system.
10,000 first visit to a specialist and then depending on what they do we tip or not more or less.
For something big like when i had my shoulder surgery we only tipped a few visits and then gave him a lump sum after surgery, maybe the "tip" is to tip before they do anything?
My shoulder surgery costs in tips about $400.and they just looked around and closed me up without doing what they told me they were going to do for me.
Was mad and never went back to the ortho clinic again, he tried to sell me shots for $300. a pop which on my own investigation seems to only work with knees and not with shoulder issues.
( The odd thing is he once gave me the same shot and it was only an extra 13,000 F, guess he got a bit greedy on me?)
Trying to sell me junk which isn't good to put into your system anyways.
I've had my husband check online and they have private day clinics that can help me out if I have any more issues.
So far staying far away from all doctors has been good for my health!

Our neighbor who used to be in the circus in Hungary is now in her mid 60's, gained allot of weight and has knee issues, probably from all the stunts she used to do with her job.
She had her knee drained because of some fluid and she told us she tipped her ortho doc 20,000 one visit and 15,000 on another visit, seems we aren't big tippers after all.
Our elderly HU neighbor in her 80's tips 10,000 to see her family doctor.
I'm not sure what they would do if we only gave them a couple thousand forints, maybe not give you much attention on your next visit?