There are several right ON Duval Street if that's what you are looking for. Speakeasy Inn, Avalon B&B, Tropical Inn. All get pretty decent reviews.

We like the Harbor Inn, which is close to Duval, but near the Seaport.

Also right on Duval is Duval Gardens. We had a very enjoyable stay there a few years ago and wouldn't hesitate to go back.

Andrews inn

That's a trick question. Best being relative. My "Best" is Marrero's Guest Mansion.

We've stayed at the Artist House B&B a few times. A couple of short blocks from Duval, but nice and quiet. They get booked up pretty quickly.

IMO, anything within 2-4 blocks from Duval is really close, especially if you have bikes. We walk/ride everywhere and you'd be surprised how close everything is to Duval.