What is the cheapest way of transport to get from the Incheon International Airport (Seoul) to Daejeon?
Take the Airport Railroad to Seoul Station and from there a "Mugunghwa" train to Daejeon. The "all stop" train (1 hour journey) from the airport to Seoul costs 3950 KRW. There is an express train doing the journey in 30 minutes. It costs 8000 KRW.
The train from Seoul top Daejeon (journey time: 2 hours) costs 10800 KRW in economy class with an assigned seat. If you do without the reservation, you will save 1600 KRW.
In total this option (13150 KRW for its cheapest variant) is cheaper than an "Airport Limousine" (= bus), which costs 20600 KRW.
There are two slightly more expensive alternatives
There are direct KTX trains from Incheon Airport to Deajeon. The 2 hour journey costs 36000 KRW in economy class with an assigned seat.
As an alternative, from Seoul Station you can take a KTX to Daejeon. The 1 hour journey costs 23700 KRW in economy class with an assigned seat.
KTX service to Incheon Airport was stopped in 2018. Use the AREX train from Incheon Airport to Seoul station, then take the KTX from Seoul station to your destination.
I know you're asking for the cheapest, but I would still recommend the express bus direct from Incheon Airport to Daejeon:
From KRW 20,600, services every 20-40 min (in normal times, anyway, COVID has likely slashed these). This avoids a transfer at Seoul Station, which can be painful if you're carrying lots of luggage, and gets you there faster than regular trains as well.