I've typed this a thousand times, so I suppose I could just search it and copy/paste, but I'll just try to re-type. (This is my criteria for Catholic Mass; I use different guidelines for different traditions)

1. First priority is the sung propers. Can I find anything that paraphrases the proper, either the text or the message? EG, "Sing, O Sing, This Blessed Morn" for Mass of Christmas Day at entrance. If the proper is from a psalm of which I have a suitable metrical setting, I may use that also - if the proper is from Psalm 103, I may use "O Bless the Lord, My Soul". If there's nothing that fits either of those, I might select something which matches the general "theme" of the proper - repentance, mercy, praise, etc.

2. If that isn't doable, and often it isn't, I then look at the readings. Is there a text which tells the story of the readings, or uses it to expound a point? For example, "O Fili et Filiae" for Dominica in Albis. The question there is where to put it, and I generally prefer proximity if possible (although in a Catholic Mass I prefer to have hymns chiefly at the end of Mass). Hymns related to the epistle before Mass, related to the Gospel at Offertory. Again, sometimes the lessons clearly indicate a "theme". Also good to look at are the collects. At least with the Revised Common Lectionary we use at my Episcopal church, the lessons for our current season are very hard to match, but the collects just give away which hymn one should use!

3. If you still have some open "slots" left, or neither source is inspiring you, select hymns related to the season. But be careful to discern the season well! "Lo, He Comes with Clouds" is not appropriate on the 4th Sunday of Advent, and neither is "Veni, Redemptor Gentium" appropriate on the 1st Sunday! And "Lo, How a Rose" is most appropriate for the Vigil Mass of Christmas, and much, much less so for the other Masses of Christmas. Often the lectionary during the summer is heavy on Eucharistic themes.

4. As a last resort, select hymns which fit the liturgical action. For the entrance, one of the "Psalms of Ascent" is appropriate, or a hymn of "gathering" or sacrifice. At offertory, a Eucharistic hymn is appropriate. During communion, I strongly suggest refraining from hymn singing, but one may choose a hymn of thanksgiving. After Mass, a hymn of mission is good. I will caution that there is a paucity of suitable hymns along these lines. They range from heretical to horizontal to rubrical. But digging deep through the hymn treasury will produce some suitable choices along these lines - in another thread, the didache was mentioned, so why not "Father, We Thank Thee Who hast Planted"?

It goes without saying that any hymns selected will be of suitable musical and textual quality for Mass and in line with Catholic teaching. I don't even consider anything that isn't. And furthermore, the sensibilities and competencies of the congregation have to be taken into account, but that's another topic altogether!

Sorry Adam:

Gavin already went against your wishes, but he does have excellent criteria! I pretty much subscribe to the same thing.

I have been thinking about this a lot lately. One of my criteria is that Christocentric always overrides Self-o-centric texts. This eliminates the music that 'celebrates self' or 'celebrates community'. It should never be programmed for the Mass.

Then I also have the 'musical judgement' criteria which eliminates trite melodic content. For example, "Just a Closer Walk With Thee", or the sappy Marian music of the early 20th Century.

Gavin:

Is there a list of hymns that are aligned to the Proper? That would be a wonderful resource. Cantica Nova has somewhat of a list, but I don't think it sticks to aligning itself with propers as much as the readings of the day.

with some exceptions, coincide with the Catholic lectionary.

Some exceptions, yes...

Side story:
When I got my current Episcopal job several months ago, I sat down to plan a few months worth of music. Knowing that the ECUSA had adopted the Lectionary a few years ago, I was using my own print Study Edition of the Sunday Lectionary as a reference. After reading about 6 or 8 weeks worth of readings, I cam across an OT lesson from the book of Wisdom.
Wait....
I don't think Anglicans recognize the Deuterocanoncials..
oh crap...

That's when I found out that, especially during the weeks following Pentecost, the Revised Common Lectionary varies quite a bit from the Catholic Lectionary.
(And then, after some research, I had to explain to the Rector that there were TWO options within the RCL for OT and Psalm, and I need to kow which one we're doing. No, not the BCP vs. RCL readings- two different tracks within the RCL. No, we can't pick them out on a week to week basis, you do one track or the other. Yes, I need to know two months ahead of time, I'm trying to plan music....)

I use three sources:

--CanticaNova's suggestions
--the Graduale Romanum
--the scriptural cross-reference section of the current OCP Master Index (I don't have a copy of the WLP one), on the off chance I might find an appropriate hymn related to the Gregorian propers (it's a rare occurrence, but it does happen from time to time)

In the past, I've also looked at pieces from By Flowing Waters, especially for solemnities and feasts. I've held off doing so since our new administrator came in, though, at least until I've had a chance to talk with him about the parish music program. I try to avoid "praise and worship" as much as possible, although there are some modern composers whose works I have little trouble with--Lucien Deiss and Alan Hommerding are two of my favorites.

At the EF:

Greatest weight is given to traditional Catholic hymns which people already know. Almost all hymns were composed before 1962. This is wonderfully liberating, and also inexpensive. I try to use Latin for Offertory and Communion, and either vernacular or Latin for procession and recession.

1. I try to avoid hymns containing the text of Propers or paraphrases thereof at the EF. (I understand why the opposite practice might pertain at the OF.)

2. Readings are only rarely a guide. Some Sundays call loudly for hymns based on readings, like Good Shepherd Sunday and Low Sunday. For others there are few traditional Catholic hymns, in English or Latin, to choose from. Anyone ever wonder why, since the EF lectionary was stable for 1300 years?

3. Seasons - I also rely heavily on devotional months, especially May through November, and proximate major feasts, like Mary's birthday, or even minor feasts if they fall on the Sunday itself, even if not commemorated at Mass.

4. As far as liturgical action hymns, Eucharistic hymns are suitable as a last resort for any part of the Mass.

5. Finally, hymns can be programmed like bidding prayers based on events of pastoral importance. For example, we sang "Long Live the Pope" after the scandal broke earlier this year, and the response was rousing.