To beging with, we recommend going to Xampanyet (Carrer Montcada 22). Try a glass of their home made xampanyet (sparkling wine, 1,10€) or vermouth (1,30€). Their "berberechos" (cockles) are spectacular (5€ half serving).
Afterward we recommend you to have dinner at Mercat de la Princesa, a new fantastic gastronomic space where you can taste different kinds of food. Placed in a medieval house, it is a food-court with long tables to share in the middle and around it different food desks where to pick up your favourite dishes. "Huevos estrellados" (eggs mixed with all kind of food), fried tapas, pasta, rices, sausages and so on... We recommend you the black rice and the fideua as well as the "pescado in adobo" at the "freiduria".
You can reach Mercat de la Princesa from three narrow streets: Carrer Sabateret, Carrer Flassaders o Carrer L'Hostal de Sant Antoni. They are very close to Picasso Museum (Montcada st and Princesa st). Open from Sunday to Thursday until 24.00 h, Friday and Saturday until 24.30 h, and Sunday until 23.00 h.
In case it is full or you preffer something even more casual, then we recommend eating dinner at Pim Pam Burger, which is a small hamburger parlour in the heart of the Borne. Their hamburgers are great, but the French fries are arguably the best in the city. 10€ per person. Carrer Sabateret, between Carrer del Rec and Carrer dels Flassaders (two of the streets that branch off from Passeig del Born).
Another option is La Paradeta del Borne, a very original restaurant that simulates a fish stand in the market. When you arrive, you select the fish and the way you'd like it prepared (fried or grilled). You then pick up your order yourself and dig in! The fish is on display in the entrance. Choose whatever you like! Market price by weight. Around 20€ per person with vino turbio (unfiltered table wine) included. Carrer Comercial 7, behind Mercado del Borne.
If you feel like drinking something after dinner, we recommend cocktails in "El Born" (Passeig del Borne corner of Calders). Go upstairs to the lounge and kick back on the sofas and armchairs.
To get home, you can catch metro line 4 at the Barceloneta station, or take a taxi for around 5-7€.