There is a train from Bucharest to Istanbul every day at 12:20 pm, it merges with the train from Sofia in Eastern Bulgaria and arrives in Istanbul at 7:30 the next morning. One source mentions 36 Euros as the price.
There are also buses going directly from Bucharest to Istanbul taking 18 hours.
I am not sure about the prices but I travelled on trains and buses in Romania and Bulgaria last autumn and there were no significant differences between the two. Considering your friend may already take a bus from Istanbul to Georgia, I would recommend the train from Bucharest, personally I would even take the train at least to Ankara and then a bus from there:
Bucharest -> Istanbul by train (36 Euros, 19hr), spending the day in old Istanbul, in the evening taking a ferry (1 Euro, 25min) to the Asian side to board the night train to Ankara (20 Euros, 9hr). From there take a bus to Trabzon in North Eastern Turkey (20 Euros, 13hr). Finally another bus to Tbilisi (14 Euros, 20hr). This may not be the fastest or cheapest but it breaks up a long journey into smaller parts.
Prices and times are from various sources and may be higher now.