A small budget is actually an advantage. The best cheap cycling gifts are the things riders use up, lose or never get round to replacing.
Buy the consumables. Chain lube, tyre levers, inner tubes and chamois cream all run out. Nobody minds a spare, and there’s no risk of getting the “wrong” one.
Socks are the safe clothing gift. Cycling socks go by shoe size and stretch, unlike jerseys and gloves, which run small and vary between brands.
Lights need no fitting. A small USB rechargeable light clips onto almost any bike. It’s a backup rather than a main light, but every commuter is glad of one when the clocks have gone back.
Check the bike if you can. If they leave it in the office, look at the handlebars. Drop bars mean bar tape is fair game. Flat bars mean it isn’t.
Make it funny, keep it useful. Chamois cream or a turbo towel gets the laugh at the office party and still gets used.