Dog limping with swollen paw: What's wrong and how can I help at home?

That swelling may indicate something stuck under the skin and causing severe inflammation (like a piece of glass or a grass seed) or something like a broken toe. Less common would be something like a sprain. At home you need to restrict Olivia's activity so no walks, running or jumping. You also need to prevent any licking by having her wear an elizabethan collar.
Unfortunately, there may not be much else you can do at home. You can apply a cool (not cold, just cool) compress 3 times a day for 5-10 minutes. If there is an infection or a break, this requires medical care. I recommend to play it safe and have Olivia examined by the veterinarian. Certainly, if she is continuing to limp after 2 days of rest or if she is getting worse, it's time to have her examined. This is the best course of action to accurately diagnose, treat and to make her feel better. I hope this helps and please feel free to post any additional questions.