I definitely think "time spent" in this situation would be the prefect gift. Is it possible to have a sitter for you other 2 children and do a special mom and daughter day with your birthday girl? Not to exclude your other kids to be mean or anything, but if you have a close relative who would keep them for you for no cost and then you wouldn't have to spend more on them coming along for your daughters day. And it could be more personal to her.
Maybe take her for ice cream. Go to a pet shop or store. Window shop. Take her to have her picture taken to remember the day together. A lot of places are running inexpensive packages right now around the holidays with no sitting fee and only like 10 bucks for a small package. Or else ask a kind stranger to get a nice picture of the 2 of you while you are out and frame it for her in a cute frame from the Dollar Store.
Speaking of Dollar Store/ Dollar Tree... They have a lot of cool, girly stuff. Depending on what you want to do, they have feather boas, tierras, dressy type things... you could get on her level and wear the stuff with her and run around and be a kid for the day with her.
You could go to the library and read books if she enjoys that. Let her get her own library card and bring books home. That would be free. Check out your local advertisements for what is going on in your area. A show at a local school or theater might not be too high in price.
Stay home and make a cake together. Allowing her to help with the baking and decorating... most girls her age that i know LOVE to help in the kitchen. For me (my daughter is only 3 and I only have one child) but I know she would really appreciate if I took a time out and just let her help me. In today's world we are always so busy and rushed in our daily lives. Even if i take 15 minutes out of our trip to Target and just take her up and down the toy isles and let her touch and look at and try out any toy she wants, she LOVES it. A dollar pop corn bag there. Just the little things seem to mean the most to them at that age.
Is there a park, river, or pond nearby? I know it's cold, but you could dress warm and take some bread to feed the ducks and geese. Make sandwiches at home if you can't afford a lunch out, and spread out a blanket and have a fancy picnic/tea party on the floor. Let your daughter drink from a fancy cup, ect.
I hope this helps and I hope you get a lot of suggestions and have a fun time with your daughter!