What does 融 in 融资 mean?
It is true that, generally speaking, 融 means to melt, but has other meanings too, especially in the economic context.

In this case, as 賈可 Jacky wrote, the character 融 can be translated as circulation or to circulate. In the online Chinese Dictionary zdic this character is defined as follows:


  ● 融
  róng  ㄖㄨㄥˊ
  
  
   固体受热变软或化为流体：～化。～解。消～。
   调合，和谐：～合。～洽。～汇贯通。其乐～～。
   流通：～泄（飘动，浮动）。金～（货币的流通，即储蓄，信贷、汇兑、股票和证券交易等经济活动的总称）。
   长远、永久：～裔（形容声调悠长）。
  


The meaning we are interested in is #3, which says:


  [...]3. Circulation [流通]: like in the words fluctuation [融泄] (floating, drifting) and finance [金融] (circulation of money, namely savings, credit, remittances, buying and selling of stocks, and other economic activities of this kind)[...]


So, this character 融 has to do with circulation (流通) of capital (资本), and that's why the meaning of 融资 is in fact financing or funding.

This particular meaning of 融 can also be found in the 《汉语大字典》 and in the 《辞源》. Both dictionaries explain this character by saying that its meaning is basically 流通 (to circulate).

I couldn't find any reference to the ancient tradition your friend mentioned. Anyway, it seems to me that the word 融资 is quite a modern coinage.

References:


融资 entry in Wiktionary (Traditional characters)
融 entry in Wiktionary
融 entry in zdic