I used to have an old LCD 19" desktop monitor which would KILL me with headaches.

Recently I got a 24" LED 1080p Dell Monitor that I USE via HDMI cord, got it set to 1080p resolution at 60hz, even have the brightness turned down...but after a few mins of using it my head feels all DIZZY and hurting.

However, when I use laptops I don't have any problems at all - when I use my dell monitor with my xbox, I don't have any problem either (though I'm much further away) - i even try leaning back in my chair to make as much distance as possible when i use my dektop and it still hurts

is this a common problem? should I just get a gaming laptop? My gaming desktop rig is killer, but my headaches are unbearable!

One thing I will mention, I don't get headaches when I use the cheap small monitors you see in libraries and schools, so should I maybe try to get a smaller screen monitor? do they make 1080p hdmi monitors with small screens?Your problem might be called an 'ergonomic issue' rather than a hardware issue.
A few relavant links:
http://www.ergo.human.cornell.edu/ergoguide.html
http://ergonomics.about.com/od/office/a/laptopasdesktop.htm
http://physicaltherapy.rehabedge.com/tm.aspx?m=59226&mpage=1&key=ergonomic&

The above are not quick solutions you your problem. They do represent the vbrad scope of the issue.
User discomfort, such as headaches,  can be cause by the DISPLAY and the ambient lighting as well as the display size. And as we get older, out eyes can not focus well at short distances, except in a well lighted room. Also, back strain and wrist strain  can contribute to fatigue.

In answer to your question, Yes. There are good old small 15 inch  monitors with high resolution. In fact, a local computer dealer may have one you will sell for $25 just to get rid of it.

You'd really need to work out what is causing the headaches - It could be a number of different things.  It may or may not help but you could look into F.lux - It changes all the colours on your screen to a more reddish tone depending on the time of day - You stop noticing it very quickly but it makes the image a lot easier on the eyes. http://justgetflux.com/I pushed my monitor ALL the way to the back of the table and pulled my keyboard and mouse to the front edge of the table.

so now theres about 3 ft distance between me and the monitor rather than 1 ft previously -  hope this helps!You can as well then get a wireless keyboard mouse and sit away from the table. So that there is more distance between you and the screen some computers have a setting you can use that may help get rid of your headaches it was designed for people with epilepsy or photo-sensitive seizure warnings.... go under your computers ease of accss menu in the control panel (not sure but i think it would be under that category). what it does i'm not sure but i think it lessens the intensity of the colors on the screen without messing with the brightness.I have had very bad experiences with dell monitors too.  I got a new dell wide screen at work a couple of years ago and had to go back to my old one.  I would get a headache in a few minutes no matter how I adjusted the settings.  I use samsung and a toshiba and no problems. 

Today I got a new monitor at my new job  Dell wide screen  Within an hour I had a splitting headache.

I hate dell monitorsThis is now an old thread.
FYI; The evidence points more to the type  of monitor ,not  the brand name. But the research is not yet absolute. Some people are more susceptible than others. Of interest is the term 'flicker rate' as to how it affects  human vision.

We FIND absolute contradictions made by so-called 'experts'.
http://www.flatpanelshd.com/focus.php?subaction=showfull&id=1362457985
Which claims:
Quote

PWM is not the problem. The problem is the way PWM has been implemented in modern LCD monitors with LEDs, as it can cause eyestrain...

The flat-screen LCD monitors have no flicker rate, so they don't trigger Migraines. The prices of them have come way down the last few years. With an LCD monitor, you should be safe from triggers unless you come across a SITE that still has one of those awful flashing ads.
... Teri Robert
Lead Health Guide, Health Central Migraine Community