Please forgive me if there are caveats or circumstances to this idea that I am not aware of.
The "good work" that one can do by eating fish on Fridays is penance. Perhaps it's hard to understand if you love fish and abhor meat, but that's not the case for most people. The following lines contain an excerpt of Keeping Friday, from the Canadian Conference of Catholic Bishops:
Jesus invited to carry the Cross and do penance for the good of the Church and the world. He died on Good Friday and, from that moment on, we share in the Lord’s Cross each Friday, learning to obey with Him, joining in His death to sin and evil. Every Friday is a promise from God that we who are baptized will rise again with Christ. We suffer and carry our crosses now, but we will share in Jesus’ glory if we are faithful by his grace.
How can we do this penance on Fridays? We may do it through these actions:
- Abstinence from meat or some other form of food, drink or entertainment [Note 1: this is where the "eating fish" custom comes from; you exchange one for the other].
- Prayer: take part in a service of worship with others, or pray with our family, or spend some extra time in personal prayer.
- Good works [Note 2: meaning concrete exercises of charity, because abstinence and prayer are also "good" works]: do good to others by visiting the sick or aged, helping those in any need, or by contributing time or money to a work of charity.
From the Catechism of the Catholic Church:
1438 The seasons and days of penance in the course of the liturgical year (Lent, and each Friday in memory of the death of the Lord) are intense moments of the Church’s penitential practice.36 These times are particularly appropriate for spiritual exercises, penitential liturgies, pilgrimages as signs of penance, voluntary self-denial such as fasting and almsgiving, and fraternal sharing (charitable and missionary works). (540, 2043)
The time during which Catholics are supposed to abstain from meat is on Fridays during Lent. We do this in memory of Christ's death. The eating of fish stems from this, but the real point is that Catholics are supposed to abstain from meat. They don't have to eat fish but are supposed to fast on Fridays.
From the Code of Canon Law:
Days of Penance
Can. 1249 The divine law binds all the Christian faithful to do penance each in his or her own way. In order for all to be united among themselves by some common observance of penance, however, penitential days are prescribed on which the Christian faithful devote themselves in a special way to prayer, perform works of piety and charity, and deny themselves by fulfilling their own obligations more faithfully and especially by observing fast and abstinence, according to the norm of the following canons.
Can. 1250 The penitential days and times in the universal Church are every Friday of the whole year and the season of Lent.
Can. 1251 Abstinence from meat, or from some other food as determined by the Episcopal Conference, is to be observed on all Fridays, unless a solemnity should fall on a Friday. Abstinence and fasting are to be observed on Ash Wednesday and Good Friday.
Can. 1252 The law of abstinence binds those who have completed their fourteenth year. The law of fasting binds those who have attained their majority, until the beginning of their sixtieth year. Pastors of souls and parents are to ensure that even those who by reason of their age are not bound by the law of fasting and abstinence, are taught the true meaning of penance.
Can. 1253 The conference of bishops can determine more precisely the observance of fast and abstinence as well as substitute other forms of penance, especially works of charity and exercises of piety, in whole or in part, for abstinence and fast.