Lots of answers that get part of the chemistry and metallurgy right, but that also either miss key features or are wrong in some conclusions (some facts are right, but the reasons for those facts lead to wrong conclusions). This won’t be exhaustive since I haven’t worked with mercury much, and it’s not in my home library, but this will fill in many of the gaps in other answers.
Quenching in oil versus water leads to lowered hardness (as other answers indicate), but contrary to some answers (and in alignment with others) this is neither good nor bad specifically.
Hardness of steel will alter how the metal interacts with other substances: if you hit a softer object you will cut it (in general), but increased hardness increases the risk of both chipping and breaking.
Other answers indicate that Japanese steel used differential hardening to get a hard edge and flexible backing, which while true, was because of the poor quality of iron ore and metallurgy throughout the Tokugawa period, which wasn’t significantly improved until the Meiji Restoration in the late 1800s, by which time sword development had largely ceased. While very good steels can be obtained from Japan today, it’s not because of their inherently good ores, but despite them. The opposite was true in Spain and Sweden, which had naturally occurring high quality ores that lead to superior steels with the same processing. This is a large part of why arms and armor development was so different between Western Europe and Japan. With modern steels differential hardening as seen in Japan adds negligible benefits.
One comment suggested that water was less effective for hardening because of the rate of absorption. This is incorrect: waters primary drawback is due to hydrogen embrittlement caused by the absorption of elemental hydrogen into the steel matrix, which weakens it in every regard. For maximum hardness you would want to use a high thermal conductivity liquid to quickly freeze the matrix into place forming a higher magnetite to austenite ratio.
Depending on desired features, a “best” method would be edge hardening with a liquid that doesn’t contain hydrogen and with a high thermal conductivity (pure magnetite) paired with a spine of softer spring steel (much more austenite). Assuming that mercury (or dragons blood) doesn’t form an alloy (metallurgists, please chime in if you know).
Also keep in mind that there is no “best” steel, everything is a trade off. For example, if I had to make my own sword from scratch, there’s no way I’d try any kind of laminated or pattern welded blade, but just get a nice Swedish or Spanish ore and forge a monosteel blade and be happy with it as is. The extra steps would be better, but not worth the cost. If you’re interested, different oils have different thermal conductivity, and will make different hardness blades, but remember that the real reason to use oil instead of water is to avoid hydrogen embrittlement, not slow cooling, as you can always address that through annealing (which is required under all circumstances anyway).