How did they come up with the name of logarithm?

It comes from the Greek words 'logos' and 'arithmes.' The second word means 'number'. The first word has a broad range of meanings, starting from 'word' or 'speech' (thus our word 'dialog'), going into 'thought' and 'reason' (thus 'logical'), and on to 'proportion'. In this range of meanings, it resembles the Latin 'ratio', which means both 'reason' and 'proportion' (thus the two meanings of 'rational': reasonable, and expressable as a ratio or proportion of two integers).

Thus, a 'logarithm' is a 'proportion number'. I suppose the thinking was that logarithms help in the calculation of proportions. Instead of multiplying a number by a fraction, you just add or subtract the logarithm of the fraction to the logarithm of the number.