You might take the plant or a couple of leaves to a reputable nursery to I.D. A city-run park or garden center probably has knowledgeable staff who might recognize the plant. Our county extension service is connected to a university horticultural dept and they are very helpful. You may have a phone book listing for something similar. Best wishes.

This is what the bloom looks like:

Looks like a 'Celia Red' which is a double blooming cultivar. It can be found at several Colorspot supplied retailer locations throughout the country. It could be floppy because it needs more branch wood to support such heavy blooms. Doubles blooms less frequently than singles and takes more energy away from the plant. I know everyone wants blooms as soon as they can have one but to get it healthy clipping off blooms and growing leaves might be something to consider. Give it lots of sunshine and regular watering without waterlogging it. Keep it well fed and find a way to give it as much direct sunlight as you can over the coming winter. This will help get you off to a great year of blooms next year.
Just some thoughts about how I would handle recouping a plant I gifted to someone and what might be best for the plant to get it back to a healthy and vitally blooming state again for your sister.
Happy Growing David

Thank you so much for the info. What should I feed it & how many times? Once a month? 6 months?

Try and find hibiscus food or at least a 3-1-2 ratio fertilizer. Hibs are heavy feeders and want feeding of soluable fert (like miracle gro)every 10 days to 2 weeks. If you get a granular every 3 weeks to once a month until the weather start cooling. Here in Texas I feed this kind of schedule until Halloween but you will need to slow sooner. I think it is too late for you to use osmocote which is what I use for a base fert then supplement with carl pooles hibiscus food. Green Diamond also put out a good fert for hibs. Hope this helps some.
Good Luck and Happy Growing David

Another newbie here. I purchased two "Peggy Hendri" Hibiscus the beginning of June. They are now about 3 feet tall and one has much larger buds than the other. I am an absolue novice at growing plants. After they flower, how do I prune them to keep them at a manageable size? Any other advice as to care and feeding would be appreciated

Alvibes, is your hib hardy or tropical? Toni

Sounds like a nice double yellow.(I just googled the name:) I would think the one is closer to bloom than the smaller budded one. Read through the last few pages in this forum and then if you have additional specific questions ask away.
Good Luck and Happy Growing David
P.S. Your general location is always helpful to anyone trying to help offer advise. I see zone 9 but there is a lot of zone 9 and different locations may require special care to do the best for your babies:)