I'm afraid I dont know much about homeopathic remedies so cant really recommend any.
If you want to try a herbal remedy we use these on our two (DD is almost 1yr and DS is 2+half years). Get Bioforce brand from a health shop and use "1 drop per year of age" three times a day. so for a one year old you use 1 drop. add to water, juice, squash or milk.
Echinacea...to boost immune system, recover from or prevent colds and flu.
Plantago...for anything to do with ear/nose/throat, glue ear, runny nose, blocked nose...where there is loads of mucus/snot/phlegm.
Childrens complex contains both echinacea and plantago. our two have just had a really bad cold, lots of runny nose. we used this and they improved really quickly.
Ivy/thyme...for any chesty problems, thins out phlegm and makes it easier to bring up, improves cough.
all of these are fine for a 1 yr old. (i own a health shop and have a BSc in Nutrition...just to reassure you!!)
spod, Happy birthday to your DD, our dd is one on the 23rd!
I will no longer have internet access from 18 oct, so if you have any questions, ask this week!