In general, asking whether or not all Jacobi fields on a minimal surface can be "integrated" to find a nearby minimal surface is a very difficult problem. For example, see Yau's remark here (page 246):

Unfortunately minimal submanifolds are defined by a second-order elliptic system and it is difficult to understand the deformation theory. (Given a Jacobi field on a minimal submanifold, can we find a deformation by a family of minimal submanifolds along the field?)

I think that http://arxiv.org/pdf/0709.1417v2.pdf provides an example of a branched minimal $S^2$ in $S^4$ with a non-integrable Jacobi field (see Theorem 4.1 and the subsequent comments). I'm not sure if there is a known example of a codimension one, embedded, minimal surface with non-integrable Jacobi fields.

As I'm sure you know, the converse of your question is true: any "nearby" minimal surface corresponds to a Jacobi field.

It depends on your exact problem, but you may get some mileage out of the "natural constraint," which allows you to associate a Jacobi field to a nearby surface which is minimal up to a finite dimensional error. This is described nicely in Leon Simon's book "Energy Minimizing Maps," Ch. 3.12 (or in many other places).

See also http://www.ugr.es/~aros/icm-ros.pdf, Ch 7, for a discussion of the moduli space of minimal surfaces in $\mathbb{R}^3$ with finite total curvature. I think that it is not known whether or not this moduli space will be smooth (i.e. if non-integrable Jacobi fields exist).

It is known, however, that particular minimal surfaces have no non-integrable Jacobi fields. For example the Costa--Hoffman--Meeks surfaces of all genus have no non-integrable Jacobi fields (also called "non-degenerate"): See http://arxiv.org/pdf/0806.1836.pdf.

I'll also remark that your question is also related to the following question (of Yau, I think): Does there exist a $1$-parmeter family of non-isometric minimal surfaces in $\mathbb{S}^3$? An obvious strategy is to try to rule out non-trivial Jacobi fields, but this has not been successfully carried out.

In general, asking whether or not all Jacobi fields on a minimal surface can be "integrated" to find a nearby minimal surface is a very difficult problem. For example, see Yau's remark here (page 246):

Unfortunately minimal submanifolds are defined by a second-order elliptic system and it is difficult to understand the deformation theory. (Given a Jacobi field on a minimal submanifold, can we find a deformation by a family of minimal submanifolds along the field?)

I think that http://arxiv.org/pdf/0709.1417v2.pdf provides an example of a branched minimal $S^2$ in $S^4$ with a non-integrable Jacobi field (see Theorem 4.1 and the subsequent comments). I'm not sure if there is a known example of a codimension one, embedded, minimal surface with non-integrable Jacobi fields.

As I'm sure you know, the converse of your question is true: any "nearby" minimal surface corresponds to a Jacobi field.

It depends on your exact problem, but you may get some mileage out of the "natural constraint," which allows you to associate a Jacobi field to a nearby surface which is minimal up to a finite dimensional error. This is described nicely in Leon Simon's book "Energy Minimizing Maps," Ch. 3.12 (or in many other places).

See also http://www.ugr.es/~aros/icm-ros.pdf, Ch 7, for a discussion of the moduli space of minimal surfaces in $\mathbb{R}^3$ with finite total curvature. I think that it is not known whether or not this moduli space will be smooth (i.e. if non-integrable Jacobi fields exist).

It is known, however, that particular minimal surfaces have no non-integrable Jacobi fields. For example the Costa--Hoffman--Meeks surfaces of all genus have no non-integrable Jacobi fields: See http://arxiv.org/pdf/0806.1836.pdf.

I'll also remark that your question is also related to the following question (of Yau, I think): Does there exist a $1$-parmeter family of non-isometric minimal surfaces in $\mathbb{S}^3$? An obvious strategy is to try to rule out non-trivial Jacobi fields, but this has not been successfully carried out.