Connection time in Frankfurt

If this is all one ticket no need to reclaim luggage it will be checked through to BUD

Once you arrive, you can check to see if your onward flight is available at the fast lane security screening point.

https://www.frankfurt-airport.com/en/travel/services.detail.suffix.html/article/travel/services-a-z/easy-travel/fast-lanes-.html

Frankfurt is a very efficient airport. Heck, Germany are usually very efficient.

Changing terminals, you many have to go through security again. Suggest you tell the security people that your flight leaves in XX number of minutes. They will put you to the front of the line.

Is that enough time to go through passport control, claim/recheck our luggage, and make the connecting flight

As I read this , it appears OP has 2 separate tickets. If so, even a little hiccup and its on to "Plan B".

If it is a single booking, then there should be no need to go to baggage claim. And its up to the airline to get you on to your destination, on their dime.

How the trip is ticketed is the key.

I believe I am ticketed on just 1 ticket. I just wasn't sure if I I had to reclaim baggage when I landed to go through security.

Should be enough time, and it if isn't Lufthansa will take care of you. We once made on ongoing LH flight that left at 9:05 while touching down at 8:40 due to our first flight having been delayed. Luckily it was to Istanbul so we did not have to get through immigration. But you will make it.

It's unlikely you'll arrive at exactly 8:50. According to flightstats.com, the average delay for this flight has been 36 minutes over the past two months.

If you are all on one ticket, you should not have to reclaim your luggage at FRA. When you check in at IAH, make sure your bags are tagged BUD. If they are, you just follow signs to connecting flights; you probably won't even have to go through security.

Once when changing at FRA, I mistakenly got on the wrong down escalator; instead of going to my next flight gate, I was sent to baggage claim, so I had to go back upstairs and go through security. I still made my connection, but I recommend being sure you get on the correct escalator!

Hi! I'm flying Luft 441 IAH-FRA, arriving at 8:50. I'm connecting to
Luft 1336 FRA-BUD, which has a departure time of 10:10. Is that enough
time to go through passport control, claim/recheck our luggage, and
make the connecting flight? It's just 1 hour, 20 minutes, and I
usually like to have 2 hours minimum.

You answered this yourself when you said you prefer a 2 hour minimum layover time.

This is too close even if everything works out perfectly which it rarely does and, worse, you're not traveling with carry-on only! Your inbound from IAH could easily be delayed.

Verify if your itinerary is all on one ticket and if it is, hope that the next flight to BUD is not oversold.

I believe I am ticketed on just 1 ticket.

You need to know with certainty what you have.

The difference is important.