If you change the interval to $[-1,1]$ instead of $[0,1]$, the equivalent inner product would be $$\tag{$*$} (f,g) = (f,g)_{L^2([-1,1])} + \lambda (f',f')_{L^2([-1,1])},$$$$\tag{$*$} (f,g) = (f,g)_{L^2([-1,1])} + \lambda (f',g')_{L^2([-1,1])},$$ where $\lambda = 2$.
One way to get an orthogonal basis is to start with polynomials and apply Gram-Schmidt orthogonalization to them. You get what might be called Sobolev-Legendre polynomials. If you search for Sobolev orthogonal polynomials you find lots of references, of which [MX] seems to be a reasonable review. Digging a bit deeper, one finds [M, Thm.3.3] an explicit recurrence relation for Sobolev-Legendre polynomials $S^\lambda_n(x)$, normalized by $S^\lambda_n(1) = 1$, with respect to the inner product $(*)$:
$$ S^\lambda_n(x) = S^\lambda_{n-2} + a_n (P_n(x) - P_{n-2}(x)) , $$ where $P_n$ are the usual Legendre polynomials and $$ a_n = \sum_{k=0}^{[\frac{n-1}{2}]} \left(\frac{\lambda}{4}\right)^k \frac{1}{(2n)!} \frac{(n+2k-1)!}{(n-2k-1)!} $$
[MX] Francisco Marcellán and Yuan Xu, MR 3360352 On Sobolev orthogonal polynomials, Expo. Math. 33 (2015), no. 3, 308--352.
[M] H. G. Meijer, A short history of orthogonal polynomials in a Sobolev space I. The non-discrete caseA short history of orthogonal polynomials in a Sobolev space I. The non-discrete case, Niew Arch. Wisk. 14 (1996), 93--112.
