* Make a point of acknowledging January birthdays. At least send a card if you're not in the habit of sending gifts. If you send gifts, ensure they are commensurate with your usual level of birthday gifts and NOT wrapped in holiday paper. This is one time it's worth investing in a roll that says "Happy Birthday."
* Some people dislike a birthday fuss. That is fine. Do something nice for them and don't mention the birthday. Invite them out to dinner, a movie, iceskating, hitting up that foodcart with the caramel apple cider, whatever you think they'd enjoy. Just make them feel important.
* If you are budgeting, it is your responsibility to do so mindfully. You damn well know who has a January birthday. Write it down if you can't remember it otherwise, and plan accordingly.
* Consider shopping for January birthday presents at the same time you shop for winter holiday presents, or for birthdays in mid-to-late January, aim for the after-holiday sales. You can find fantastic deals on things like winter sweaters and candy that way, or toys for children. If you like regifting, hit the thrift stores for all the stuff people gave away because it didn't fit or was duplicated.
* The only time it is acceptable to combine winter holiday and birthday shopping into one bigger gift is if the recipient specifically asked for that to happen because a big-ticket item would otherwise not fit into either budget. (Same goes for November birthdays as January ones.) This is among the most common ways that January folks get cheated, and they almost universally hate it -- even if they're too polite to say it to your face. If they ask, though, it's okay: the birthday person's wishes trump all standard etiquette.
* The traditional birthstone for January is garnet. This is a suitable gift if you have seen the person wearing it; some people love collecting jewelry, scarves, ties, etc. in their birthstone or its color. Most garnets are wine red, and these are quite affordable. Hessonite is a cinnamon orange version, also affordable. The brownish phases of several types cost much less than their brighter counterparts of red, orange, or yellow. This is not only an excellent choice for people who prefer earth-toned jewelry, but for inclusion in multicolored garnet jewelry of autumn tones. Including those reds and browns moderates the price enough to afford a bit of yellow or orange. Malaya garnets run pink to peach, originally disdained but now much more popular and pricey. The vivid orange Mandarin garnets (a subtype of spessartite) are more expensive than the usual reds. Andradite is bright yellow to brown, valued for its rarity, and priced accordingly. Tsavorite garnet is a green more piercing than emerald, and ruinously expensive. Demantoid is another green variety of similar value. If the person doesn't wear garnet because they dislike the prevailing wine red, ask if they've seen the other colors. Even if you can't afford them, it's fun to look at pictures.
* Here are some fun ways to celebrate January birthdays in a season when people are often tired and broke. It's totally okay to have a small party at home rather than going out in the (usually awful) weather. You can go with the flow and celebrate with winter activities, or fight it by pretending it's summer. Consider a party theme. If you're overstuffed from holiday junk food, behold the glorious cake of fruit. There are plenty of free or cheap things to do with friends and family.
* And then there's the escape hatch: instead of celebrating in January, celebrate the antibirthday in July. Nothing says anyone's special day actually has to be their natal day.