Do non-second-countable spaces have "small" non-second-countable subspaces?

The answer is yes.

There is special terminology and known results along these lines.

If every space with property $P$ has a subspace of size (at most) $\aleph_1$ with property $P$ then one says that property $P$ reflects in size $\le\aleph_1$ spaces.

For example non-metrizability (for compact spaces) reflects in size $\le\aleph_1$ spaces, i.e. every non-metrizable (compact Hausdorff) space has a non-metrizable subspace of size $\aleph_1$ (a result of Alan Dow). (Note that in this context $\aleph_1$ and $\omega_1$ are often used interchangeably, he writes $\omega_1$ in his paper. Similarly $\aleph_0$, $\omega$ and $\omega_0$ are used interchangeably.)

Recall that $w(X)$ denotes the weight of $X$, i.e. the smallest infinite cardinality of an open base for the topology of $X$.

The answer to your question is yes, see
Having a small weight is determined by the small subspaces,
Authors: A. Hajnal and I. Juhász
Journal: Proc. Amer. Math. Soc. 79 (1980), 657-658.
http://www.ams.org/journals/proc/1980-079-04/S0002-9939-1980-0572322-2/
Abstract: We show that for every cardinal $ \kappa > \omega $ and an arbitrary topological space X if we have $ w(Y) < \kappa $ whenever $ Y \subset X$ and $ \vert Y\vert \leqslant \kappa $ then $ w(X) < \kappa $ as well. M. G. Tkačenko proved this for $ {T_3}$ spaces in [2]. We also prove an analogous statement for the $ \pi $-weight if $ \kappa $ is regular.

Here the property that "reflects" is "having weight $\ge\omega_1$".
If $w(X)\ge\omega_1$ then $w(Y)\ge\omega_1$ for some $Y\subseteq X$ with $|Y|\le\omega_1$. Indeed, if this were not the case, then taking $\kappa=\omega_1$ in the above theorem, we would have that every subspace $Y\subseteq X$ with $|Y|\le\omega_1$ satisfies $w(Y)<\omega_1$ (i.e. $w(Y)\le\omega_0$, $Y$ has countable weight, $Y$ is second-countable), and then the conclusion would be that $w(X)<\omega_1$ (i.e. $w(X)\le\omega_0$, $X$ has countable weight, $X$ is second-countable).

For reflection of non-metrizability (with a proof using elementary submodels) see:
An empty class of nonmetric spaces, Alan Dow,
Proc. Amer. Math. Soc. 104 (1988), 999-1001.
http://www.ams.org/journals/proc/1988-104-03/S0002-9939-1988-0964886-9/

And here is a more-or-less random selection about reflecting other properties
(just a few sample links, out of many more).

On Dow's reflection theorem for metrizable spaces
Jerry E. Vaughan
TOPOLOGY PROCEEDINGS Volume 22, 1997, 351-360
http://topology.auburn.edu/tp/reprints/v22/tp22123.pdf

Reflecting point-countable families, Zoltan T. Balogh
Proc. Amer. Math. Soc. 131 (2003), 1289-1296
http://www.ams.org/journals/proc/2003-131-04/S0002-9939-02-06621-2/

Reflecting Lindelöf and converging $\omega_1$-sequences
Alan Dow, Klaas Pieter Hart
arXiv:1211.2764v2 [math.GN] http://arxiv.org/abs/1211.2764
(Submitted on 12 Nov 2012 (v1), last revised 17 Jul 2013 (this version, v2))

Reflecting Lindelöfness
James E. Baumgartner, Franklin D. Tall,
Topology and its Applications
Volume 122, Issues 1–2, 16 July 2002, Pages 35–49
http://www.sciencedirect.com/science/article/pii/S0166864101001353

Ideal reflections
Paul Gartside, Sina Greenwood, and David Mcintyre
TOPOLOGY PROCEEDINGS
Volume 27, No. 2, 2003, Pages 411-427
http://topology.auburn.edu/tp/reprints/v27/tp27203.pdf