Most dayrooms only can be used from 0900h onwards; anything earlier would require you to purchase a whole night.
Just for a reference, Carat Hotel at FRA (with shuttle bus) charges 50 € for a double room (check in from 0900h, checkout until 1700h). Rates at other hotels should be similar ... (but I haven't checked rates of the hotel(s) immediately at the airport).
As I assume that you and your grandfather are on flights 3519 and 464, you only would be able to use a dayroom between 0900 and, say, 1100/1130h before having to retrun to the airport for your 1400h departure to MCO.
It really is a split decision here:
50 € for 2 persons giving them the option to sleep (if wanted) for 2 hours plus enabling them to have a shower is not a bad thing. Will make the 22-hour trip more bearable.
OTOH, if you have access to a lounge for free, you can enjoy free drinks & food there w/o having to leave the airport.
In case you would have to pay for lounge access, I probably would prefer the day room and get something to eat & drink in the airport supermarket before heading to the hotel.
Carpe diem ! Life is too short to waste your time ! Keep in mind, that today is the first day of the rest of your life !