What Gethsemane means
The Garden of Gethsemane, on the side of the Mount of Olives in Jerusalem, is where Jesus went to pray before His ordeal on the cross of Calvary.

Olive oil is pressed from the fruit of the olive tree
The Bible says that Jesus was troubled and overwhelmed with sorrow, to the point of sweating drops of blood. Three times Jesus prayed that if it were possible the cup of suffering would be taken from Him, but that the will of the Father would nevertheless be done.
Chapter 12 of the book of Hebrews tells us that Jesus “endured the cross, scorning its shame” and that He did this “for the joy set before Him”. In other words, although He was under immense pressure, and in torment at the suffering He was undergoing, He pressed on, firstly, because it was the will of the Father and, secondly, because, by faith, He knew that He would have great joy on the other side of the suffering.
The setting of His spiritual travail before the crucifixion, at Gethsemane, and even the name of the place, is deep with meaning.
Gethsemane is a corruption into English of two Hebrew words GAT and SHMANIM. As others have pointed out, it is taken to mean “the place where olive oil is pressed”.
Olive presses are to be found in Israel and throughout the Mediterranean region.
At an olive press, olives were gathered into rough sacks and stacked one on top of another. A beam was lowered onto the stack and increasing weight was added to the end of the beam to press oil from the olives. The more pressure, the more oil.
Olive oil, in the symbolism of the Bible, is a picture of the presence of the Spirit of God ( Gen 28:16-18 ). Oil was used for anointing priests ( Exo 29:7 and Lev 8:12 ), and kings ( 1 Sam 10:1 and 2 Sam 2:4 ). It was also used to bring light ( Exo 27:20 ) from the menorah in the Tabernacle in the wilderness.
Olive oil is produced under pressure. It brings joy to those who produce it ( Psa 45:7, Isa 61:1-3, and Heb 1:9 ).
Now, here is a further insight into the Hebrew words GAT and SHMANIM.
GAT, as a Hebrew word by itself (Hebrew: gimel, tav), means a winepress.
SHMANIN means oils, and is the plural of the Hebrew word SHEMEN, for oil.
A literal translation of GAT SHMANIM would be “winepress of oils”.
Adding the imagery of a winepress brings an extra dimension to the picture. The Bible says that wine “gladdens the heart”. Wine is not to be taken ‘unto drunkenness’, but it is a gift from God. There are several pictures in the Bible of wine and oil symbolising the favour of God ( Psa 104:14-15, Hos 2:8, Hos 2:21-22 and Joel 2:24 ).
The suffering and sacrifice of Jesus allowed the Holy Spirit to be poured out on those who believe in Him. There is joy in our salvation, and joy in the promise of what is to come in eternity.
Trials and tribulations, pressure and suffering, are part of the human condition. Christians do not escape.
However, Christians have the joy of knowing that God is with us through times of pressure and suffering, and that He allows it because the final outcome will be good ( James 1:2-3, and Rom 8:28 ).
