The storm is expected to hit Scotland with the most force and the agency has issued a yellow - “be aware” - severe weather warning for north and west Scotland from 6pm on Thursday till 2pm on Friday.
Abigail was chosen by the public as part of a scheme to raise awareness of severe weather to help improve safety.
It said the severe south-westerly gales risked disrupting transport and large waves battering coasts.
Gusts of 60-70mph are likely, and more than 80mph in exposed locations.
The warning area covers the Highlands and Islands, the west coast north from Islay, and northern Aberdeenshire.
A Met Office spokesman said: “A vigorous depression is expected to pass just to the north west of Scotland on Thursday night, bringing a swathe of very strong winds on its eastern and southern flanks.”
Chief meteorologist Eddy Carroll added: “There remains some uncertainty regarding the exact extent and timing of strongest winds, but current forecasts suggest this storm could have some medium impacts such as disruption to transport, and so a warning has been issued and the first storm has been named as Abigail.”
The other names chosen for storms are Barney, Clodagh, Desmond, Eva, Frank, Gertrude, Henry, Imogen, Jake, Katie, Lawrence, Mary, Nigel, Orla, Phil, Rhonda, Steve, Tegan, Vernon and Wendy.