Day trip from Budapest

Do you mean Split? If so, that is one long day trip -- it's almost 500 miles so I assume that you would fly there but I don't know of any non-stop flights to Split so that means about 4 hours each way of travel time. Vienna on the other hand is a 2 hours from Budapest by train and although it is worth several days, you could make a day trip there. Have you considered Bratislava in the Slovakia or taking a day cruise on the Danube or Esztergom in Hungary?

Good day trips are Szentendre, Vac, Esztergom, Visegrad
But it looks like you have time for an overnight trip in which case:

Eger: Great wine town with a fort on a hill above a lovely old town square
Pecs: Complete with ancient mosques and some the the nicest pedestrians in Hungary. Generally about 5F warmer than Budapest so great in the fall
Gyor: Excellent baroque old town and near the Pannonhalma Archabby / Winery. I really like this place
Tihany: On lake Balaton is one of my favorite villages. Pretty much an arts village, great old abbey church and spectacular views of the lake.

Sorry, I did drop the L from Split. 500 miles does sound like too great of a distance so I'll look at that for the next vacation. We are coming from 5 days in Vienna and plan on hitting Bratislava on the way to Budapest, so we will be taking those two suggestions. We'll take a look at Szentendre, Vac, Esztergom, Visegrad for the day trips. The idea of doing an overnight away from Budapest sounds pretty good. Pecs, Eger, and Gyor sound pretty good. I have been to Lake Balaton before and you are right the views are great. Our primary interest are in the historic type of sites and checking out the local scenery and people. My Grandfather came from NW Hungary and my Grandmother from Bratislava so looking forward to learning more on the history and culture.

Joe

After Bratislava you might want to spend a night in Gyor on your way to Budapest. Or if the refugee situation is still in full swing you might do better to head south to Pecs. there are few places in Hungary that drip in more history Roman - Ottoman - European than does Pecs.

That sounds like a very good suggestion. I'll start looking at doing that. What is better for seeing the sites. Boat or Train down to Gyor?

Train to Gyor

I would not overnight anywhere else since it sounds as if you only have five nights in Budapest. My wife and I did a day trip to Eger on our 2011 Hungary-Czech Republic trip, and even though it was a bit far we thought it was a nice side trip.

I saw that Eger talks about the wine country there. What did you see there?