Before choosing hymns for a given Sunday I reflect on the readings for that day and sketch out a list of hymns that reflect themes or messages in the bible passages. I think this is a fairly standard practice for any lectionary based parish. Some music directors are given the theme of the pastor's sermon and go from there, or maybe they are only given a liturgical season. But at any rate, the first consideration is the text. I check to make sure the texts of all the music, including the choral anthems, reflects the theology of my Church and supports the readings for that day.
I specifically choose melodies that I can easily sing a capella. If singing the hymn requires the support of the harmonization of the accompaniment, I do not choose it for my parish. Easy to sing melodies tend to make use of step-wise motion, small intervals otherwise, and musical sequences. I look for a small melodic range of B3 to D4, although if a hymn is truly beautiful and does not linger too low or too high, I'll use it. Children can have difficulty singing below B3 and older adults prefer lower ranges. So if you want everyone to sing, you'll need to limit the range. The exception is for much loved hymns that are motivating enough for people to make an effort to go outside their comfort zone. I'm always amazed at how loudly people sing "I Am the Bread of Life" as it breaks most of my guidelines for a good melody!
This last component is quite subjective: Depth. Does the text and melody combine to create a hymn that has beauty and inspires awe, joy, reverence. If the hymn inspires a feeling that it is "cute" it does not make the cut. I avoid the trite and the trivial whenever possible.
It would be nice if I could just list out songs and say, "These are the ones that work!" But the selection of music is so very specific to the congregation under consideration that it's just not possible. So much depends on the past history of the parish, the theology of the denomination, and the backgrounds of the participants. So all I can offer on this front are the broad, subjective criteria I use for my own purposes.