We just went to WDW for our first time in December and spent FIVE days there (and that felt like "barely enough" for us)!! It was me, DH, my MIL, and DD (she was just turning 5).
With just 1 day and 4 kids, I'd think Hollywood Studios would be better. You could spend a whole day there and they would really enjoy meeting so many characters. It wasn't as crowded as the other parks, but there was fun stuff to do around every corner. Plenty of opportunities to sit and enjoy short shows, too.
We LOVED Magic Kingdom, but you just won't get to see/do much in 1 day! We spent 3 days there and it wasn't until the 3rd day that I felt like I knew my way around and got to really see & enjoy things! Of course, there are a TON more rides, but with 4 kids, I think it would be more stressful trying to drag them from one end of the park to the other trying to see as much as possible.
Animal Kingdom was also really nice! We saw the whole park in 1 day and didn't feel rushed or too crowded. There are more younger-kid-friendly rides than Hollywood Studios and it wasn't very crowded.
We didn't do Epcot, but mostly because I had read the reviews about it being geared towards older kids & adults and we had a limited time.
If you could squeeze out at least 2 days (or 3), I would very much suggest MK!! For just 1 day, I would suggest Hollywood Studios (or Animal Kingdom if your kids like animal-related stuff).
I'm not sure when we'll go back, but it was an amazing trip for our family and we really enjoyed ourselves! We have friends who go every 1-2 years, but I can't imagine fronting that cost for such a "commercial" vacation!
Whatever you choose, I'm sure your family will have a blast!