Hi Rika,
I thing that clothes are not the best, because kids grow out of them too fast. But I think, that bibs are a great idea! We got national bibs for my boy from England (with the flag and his name on it), from Spain, Croatia, Austria, and we still use them with my younger son and they always make me remember of those sweet friends! Maybe a newborn won't use it in the first months, but then they'll use it for a long time!
I think you can find these in markets at Vörösmarty tér for example, but I also just found a website, where they make you bib with a photo, so you could put a Hungarian flag or picture of Budapest on it plust the baby's name. Check it here:
http://www.sooters.hu/ajandektargyak/baba-partedli?gclid=CKzV7L_D-LoCFQlb3godFwYAIA