A "second opinion" is an opinion rendered by someone who was not the original person who handled a certain matter. For example, a patient who is diagnosed with cancer and given a recommendation about treatment might seek a second opinion from a different cancer doctor regarding other treatments. Similarly, a dog attack victim whose lawyer obtained a settlement offer might seek a second opinion from a different attorney as to whether the offer should be accepted.
The attorney who is retained for a second opinion would not become your "attorney of record" in court -- he or she would not be filing court documents and otherwise protecting your rights -- but would only review your medical records and other evidence and then give you an opinion as to the value of your case. This is a difficult task because it would require this attorney to review a lot of documents pertaining to the lawsuit itself, such as the pleadings, motions, expert witness disclosures, interrogatories, requests for production of evidence, admissions, deposition transcripts and possibly other things. It's not just difficult but also time-consuming.
Since the lawyer giving the second opinion would not be entitled to a share of the settlement or judgment, you would have to pay hourly fees or a flat fee for this work. The more experience the lawyer has, the higher the cost and the faster and more reliable the results. A second opinion from a highly experienced expert can be really valuable, not only to give you a better idea about what your dog bite case is worth but also to provide you with peace of mind.