From Longman Dictionary of Contemporary Englishput somebody down for something phrasal verb1 WRITEto put someone’s name on a list so that they can take part in an activity, join an organization etc They put themselves down for a training course.2 put somebody down for £5/£20 etc especially British EnglishWRITE to write someone’s name on a list with an amount of money that they have promised to give → put→ See Verb table
Examples from the Corpus
put down for • When Miles and Evan were little, John and I were very consistent in putting them down for an hour-long afternoon nap.• You've put me down for holidays in September and I've been telling you about the cruise in October for months.• I couldn't wait to put my name down for the Swimming With Dolphins Expedition at our next port of call.• If so I'd like to put my name down for West Ham's lot, they're a bit tasty. 