Hooded Merganser: Reported by Will Jaremko-Wright and Steven Miller on January 24 and January 26. It is a male and was seen in Taos, NM. Hooded Mergansers can be found in Arizona, Texas, and Colorado, but not usually New Mexico.
Lesser Yellowlegs: Reported by Bob Nieman on January 24. Two were seen at Bitter Lake NWR in Roswell, NM. Lesser Yellowlegs are found during migration in New Mexico but not during the winter.
Thayer's Gull: Reported by Bob Nieman on January 24. It is a first cycle gull and was seen in the Mall Parking Lot in Carlsbad, NM. Thayer's Gulls can be found in high northern Canada and the Pacific coast but not in New Mexico.
Eastern Towhee: Reported by Bob Nieman, Nancy Hetrick, Wyatt Egelhoff, Patty Lovekin, and John Drummond on January 24, January 26, and January 31. There has been one seen many times at Bitter Lake NWR in Roswell, NM. Eastern Towhees usually only make it as far west as eastern Texas.
Snowy Egret: Reported by Jay Wilbur, Wayne Treers, and Judy Wilbur on January 25, January 26, and January 30. It was seen in Sunland Park, NM and Las Palomas, NM. Snowy Egrets are seen in summer and during migration in New Mexico but not during the winter.
Lincoln's Sparrow: Reported by Kaia Colestock, Matthew York, and Will Jaremko Wright on January 25 and January 29. One was seen at Bandelier National Monument, NM on the 25th and 7 were seen in Las Vegas, NM on the 29th. Lincoln's Sparrows are seen in summer and during migration in New Mexico but not during the winter.
Cinnamon Teal: Reported by Nancy Hetrick, Wyatt Egelhoff, John Drummond, Patty Lovekin, and Corry Clinton on January 26 and January 31. There were four males and one female, and they were seen at Bitter Lake NWR in Roswell, NM. Cinnamon Teals can be seen in New Mexico during the summer but not usually in winter.
Barrow's Goldeneye: Reported by Jeff Szabat, Bernard Foy, Thomas Jackman, and John 'Lefty' Arnold on January 26 and January 28. The goldeneye/s were seen Ohkay Owingeh, NM and Taos, NM. Barrow's Goldeneyes are usually seen in parts of western Canada, southern Alaska, and some parts of the lower 48 but not New Mexico.
Field Sparrow: Reported by Bernard Morris on January 26. The sparrow was seen in Lordsburg, NM. Field Sparrows usually only make it as far west as western Texas.
Northern Cardinal: Reported by Wyatt Egelhoff and Nancy Hetrick on January 26. It was seen in Melrose Woods in Roosevelt County, NM. Northern Cardinals are usually eastern birds but do make it into very southern New Mexico and most of Arizona and Mexico. They are, however, rare in this location.
Mew Gull: Reported by Bernard Morris on January 27. It was seen in Elephant Butte Lake State Park in Sierra County, NM. Mew Gulls are usually seen in western Canada, Alaska, and the Pacific coast.
Dusky Flycatcher: Reported by Brandon Miller on January 27. It was seen in Leasburg Dam State Park in Las Cruces, NM. Dusky Flycatchers are usually seen in New Mexico during migration and summer, but not winter.
Winter Wren: Reported by Brandon Miller and Will Jaremko-Wright on January 27, January 28, and January 29 (reported twice by Will Jaremko-Wright). It was seen in Leasburg Dam State Park in Las Cruces, NM and Gallinas River in San Miguel County, NM. Winter Wrens are usually seen across the eastern United States and Canada, making it as far west as western Texas.
American Tree Sparrow: Reported by Bill Maley, Lucas Laeser, and James Loy on January 28 and January 29. There have been differing numbers being seen both at the Alameda Bridge and Open Space in Albuquerque, NM (which I visited on Thursday along with the Thursday Birders from CNMAS hoping to see the sparrow but unfortunately we didn't) and the Bandelier National Monument in Los Alamos, NM. American Tree Sparrows do make it into New Mexico, but not this part of New Mexico.
Greater Scaup: Reported by Bob Nieman on January 30. Four were seen at the WIPP (Waste Isolation Pilot Plant) in Carlsbad, NM. Greater Scaups are found in Canada and Alaska and scattered locations throughout the western and eastern U.S. They are not usually found in New Mexico, however.
Blue-winged Teal: Reported by John Drummond and Patty Lovekin on January 31. Three were seen at Bitter Lake NWR in Roswell, NM. Blue-winged Teals are found in the summer in New Mexico but not the winter.
Those are all of the rare birds to my knowledge that have been seen in New Mexico this week. All the data is through eBird, and all of these birds have been reported to eBird.