We're slated to go to an "ESPN owned and operated bowl" which leaves 17 possibilities (...well 16, since the FCS Celebration Bowl isn't really an option)
Options are (ranked loosely from worst to best in my own first-glance opinion)
Bahamas Bowl (16 December): Played in Nassau, Bahamas, this is the first bowl game of the year, with an 11:30 AM Eastern kickoff, so would be a 9:30 kickoff for Mountain Time - which is particularly bad since many still are at work on 16 December. It's matched to CUSA vs MAC, so if we end up here, we'll get one of those conferences.
Hawaii Bowl (24 December): Played in Hawaii, this bowl takes place late on Christmas Eve (8:00 PM ET, ending around 11:30 ET), so isn't likely to get much viewership/interest. It matches up MWC, CUSA and AAC teams. BYU played in the Hawaii Bowl in 2019, losing 38-34 to Hawaii. Hopefully that recent appearance keeps us from making a return visit.
Cure Bowl (16 December): Played in Orlando on the first day of Bowl Season and matches up any G5 team, drawing from AAC, CUSA, MAC, MWC, SunBelt or Army. With a 3:00 ET kickoff the Friday 8 days before Christmas, there's a real chance many will still be working. BYU has never played in this bowl.
New Mexico Bowl (17 December): Played in Albuquerque on a Saturday 8 days before Christmas with a 12:15 Mountain time kickoff, this is the 4th bowl of the year (after Bahamas, Cure, and Fenway) and matches up AAC, CUSA and MWC teams. BYU played here in 2010 beating UTEP 52-24. If we end up here, it'll likely be a similar opponent, but this one might be appealing to bowl organizers for geographic reasons.
Frisco Bowl (17 December): Played in Frisco (basically Dallas-Fort Worth), this is another G5 special, taking from any of the G5 conferences. It's a particularly late kickoff (9:15 Eastern) and BYU has never played in it before, so this seems a good bet for where BYU ends up. (This was previously the Miami Beach Bowl where BYU lost to Memphis 48-55 in 2014 with the infamous brawl, but it's a whole new location and management, so don't think that counts)
Myrtle Beach Bowl (19 December): Played in Myrtle Beach, South Carolina, this one matches up AAC, MAC or SunBelt teams and has a relatively early kickoff at 2:30 Eastern. Christmas this year is a Sunday so some may still be working this Monday before. BYU has never played in this bowl (though we obviously did play Coastal Carolina there in 2020)
Boca Raton Bowl (20 December): Played in Boca Raton, with a mishmash of conference affiliations, none of them good - AAC, CUSA, MAC, MWC, SunBelt and Army are the teams that can fill this. BYU played there in 2020, beating UCF 49-23, and I'd bet we don't come back so quickly.
Armed Forces Bowl (22 December): Played in Fort Worth, set to be a matchup between the AAC and C-USA. BYU played here in 2011, beating Tulsa 24-21. If we get it, it'll be against AAC or CUSA.
Famous Idaho Potato Bowl (20 December): Played in Boise on everyone's favorite blue turf, this game matches up MAC vs MWC. BYU played here in 2018, beating Western Michigan 49-18. Geography argues to invite BYU again, but unlikely we come back so quickly.
Camellia Bowl (27 December): Played in Montgomery, Alabama and was held on Christmas Day last year, but moved to 27 December this year. Matches up MAC and SunBelt so we'd get one of those conference opponents. BYU hasn't played in this bowl before.
Birmingham Bowl (27 December): Played in...Birmingham, set to be AAC vs SEC - but with the caveat that it's their #9 priority bowl, or lower if an SEC team (or two) makes the playoff. If we get it, it'll likely be against an AAC team. I don't think BYU has played in this bowl before.
Fenway Bowl (17 December): Played (unsurprisingly) at Fenway Park in Boston, this is the 3rd bowl game of the season and matches up ACC (or ND) against the AAC. The ACC slots teams based on best matchups and so might actually send a team to this one, giving us a chance to play a P5 opponent. Unfortunately, the game starts at 9 AM Mountain Time on a Saturday. BYU has never played in this bowl.
Gasparilla Bowl (23 December): Played in Tampa, this one has some chance of having a P5 opponent since it's affiliated with the AAC, ACC and SEC. Realistically it's a very low priority game for those P5 conferences, so the chances are better that we're a replacement for one, rather than getting to face one, but hope spring eternal. BYU hasn't played in this bowl
First Responder Bowl (27 December): Played in Dallas, this is another one that could have a P5 opponent as it draws from AAC, ACC, Big 12, and PAC12. There have even been two P5 teams matched up as recently as 2017 (when Utah beat West Virginia). BYU has never played in it.
Las Vegas Bowl (17 December): Played in Las Vegas' Allegiant Stadium, this bowl should match up a PAC12 and SEC team, but has had a number of G5 interlopers, so there's a real chance BYU could end up here against a decent P5 team. It also is on ABC in primetime (7:30 ET), so should get a good audience. BYU has played here a LOT (it used to be the MWC champs' bowl), most recently in 2015, losing to Utah 35-28. Questionable if the proximity would help BYU be selected for this bowl or if they wouldn't want a repeat, but I'd guess enough time has passed.
Texas Bowl (28 December): Played at Reliant Stadium in Houston, this is a good one, matching up the SEC and the Big 12. Unfortunately, it's also decently high priority for both of those, which means better opponent but less likely we get it. The last time it had any non-P5 representative was 2009. This is the ideal, but probably a pipe dream
Note that there is also the chance to jump into the Guaranteed Rate Bowl (27 December) with a 10:15 ET kickoff if the Big 12 or Big 10 can't fill their spots. This would be a great get, on par with the Texas Bowl. It would almost certainly match us against a P5 and also make a lot of sense for bowl organizers - geographically close, and a fanbase that wouldn't be as affected by the late kickoff. That said, as we saw last year, it's pretty unlikely that those conferences don't fill this one, given the level of priority.