Offense Up is seen in Mother, Mother 2 and Mother 3. This is a PSI Ability that raises your offense by a small amount. Ninten is the sole user of Offense Up in Mother. Paula Polestar is the only one who can use it in Mother 2, and Lucas is the only one who can use it in Mother 3. It's effects can be stacked repeatedly.
Lucas's Offense Up α is default, and uses 6 PP, but Offense Up Ω is learned at Level 36. This one takes 18 PP. Offense Up α raises the offense of only one character, while Offense Up Ω raises the offense of everyone in the party. This move will not raise your Offense very much unless you are using the Omega form. Otherwise, it shouldn't add that much, and has the same basic usage of Flint's Strengthen Up.