The word for "good" here is haplos, which can be translated as simple, single, sincere, healthy, or generous. While haplos does not occur elsewher in the NT, the cognate words haplous (same meaning) and haplotetos (sincerity, simplicity or generosity) occur ten times. In Rom 12:8, 2 Cor 8:2, 9:11, 9:13, and James 1:5, haplotetos has the meaning of generosity. In the others (2 Cor 1:12, 11:3, Eph 6:5, Col 3:22), it means sincerity or simplicity. The LXX uses the word twice to refer to generosity, in 1 Chron 29:17 and Prov 11:25. Since the surrounding context, Matthew 6:19-34, is about money, it seems better to see haplos as a reference to generosity.
The phrase "evil eye" (ophthalmos poneros) has much more certainty about its meaning, since it always means stingy or greedy (Deuteronomy 15:9, Proverbs 23:6, Proverbs 28:22, Matthew 20:15). In the OT, the Hebrew raa ayin (evil eye) is translated as selfish or stingy. (The LXX translates the phrase with baskanos, stingy, in Proverbs, but leaves it as poneuresetai ho ophthalmos sou, your eye does evil, in Deuteronomy, probably because Proverbs and Deuteronomy were translated by different translators.)
Not all gospels scholars agree exactly with this interpretation, although most mention it as a possibility. There seem to be two reasons to doubt. First, the saying has an exact parallel in Luke 11:34-36, and there the context is not about money, but rather is connected to another saying about light (although note the mention of greed in Luke 11:39). In many cases, Matthew and Luke arrange the same sayings of Jesus in different contexts, so some scholars feel that Jesus' good eye/bad eye saying was not necessarily originally about money. Second, although haplos can mean generous, the phrase haplos opthalmos is not found elsewhere, so its exact meaning cannot be confirmed. Some scholars suggest a meaning for eye here that approximates heart, or draws on Greek biological conceptions of how the eye works.
To be honest, despite the doubts of some gospels scholars, I think this one is a slam dunk. I think that the more well-known metaphorical phrase "evil eye" caused Jesus to coin a new phrase, "good eye." The phrase "good eye" already existed in Prov 22:9, but Jesus used haplos instead of agathos (Heb. tov), probably to create an image of health vs. sickness. Even if Matthew is the one who placed the saying in this context (which is not certain), it illustrates that Matthew thought that the saying was about money.
The hard part, as usual with the sayings of Jesus, is obedience. Being generous is often hindered or completely stopped by our slavery to "mammon" or our worry about our own needs.
The picture: The Sermon on the Mount, from Kurtze Postill Herrn Phillippi Melanthonis vber die Euangelia der furnemesten Fest, 1545. Courtesy of the Digital Image Archive, Pitts Theology Library, Candler School of Theology, Emory University.