There is a lot of research about this. I can recommend the books
- Engl, Heinz Werner, Martin Hanke, and Andreas Neubauer. Regularization of inverse problems. Vol. 375. Springer Science & Business Media, 1996.
- Hansen, Per Christian. Discrete inverse problems: insight and algorithms. Society for Industrial and Applied Mathematics, 2010.
The former works in Hilbert spaces while the latter in $\mathbb{R}^n$.
In short, Tikhonov regularization helps to keep the amplification of noise down, but at the expense of a bias. This can be quantified as follows:
If there is noisy data $b^\delta$ with $\|b-b^\delta\|\leq\delta$ and we define
$$
x^\dagger = A^\dagger b
$$
be the (idealized, unknown) true solution
(where $A^\dagger$ denotes the pseudo-inverse of $A$ and we assume that $b$ in in the range of $A$)
and
$$
x_\alpha^\delta = \operatorname{argmin}_x \|Ax-b^\delta\|^2 + \alpha\|x\|^2
$$
be the regularized solution.
Then one can decompose the reconstruction error as
$$
\|x_\alpha^\delta - x^\dagger\| \leq \|x_\alpha^\delta -x_\alpha\| + \|x_\alpha - x^\dagger\|
$$
where
$$
x_\alpha = \operatorname{argmin}_x \|Ax-b\|^2 + \alpha\|x\|^2.
$$
We call
$$
\|x_\alpha^\delta - x_\alpha\|
$$
the data error, which is the error that is induced by the measurement error in $b$, and
$$
\|x_\alpha-x^\dagger\|
$$
the approximation error, which is due to the approximation of the (true but ill-posed) inversion of $A$.
The decomposition into data and approximation error is similar to the bias-variance decomposition in statistics (bias being the approximation error and variance coming from the data error).
The quantities $x_\alpha^\delta$ and $x_\alpha$ are given by
$$
x_\alpha^\delta = R_\alpha b^\delta,\quad\text{and}\quad x_\alpha = R_\alpha b
$$
where
$$
R_\alpha = (A^*A + \alpha I)^{-1}A^*
$$
($A^*$ is the adjoint/transpose of $A$ and $I$ is the identity operator).
This gives us for the data error
$$
\|x_\alpha^\delta - x_\alpha\| \leq \|R_\alpha\|\|b^\delta - b\|\leq \|R_\alpha\|\delta.
$$
For the approximation error we get
$$
\|x_\alpha-x^\dagger\| = \|(R_\alpha - A^\dagger)b\| = \|(R_\alpha A - I)x^\dagger\|.
$$
Since one can show that
$$
\|R_\alpha\|\leq \tfrac1{2\sqrt{\alpha}}
$$
and
$$
\|(R_\alpha A - I)x^\dagger\| \to 0 \quad\text{for}\quad \alpha\to 0
$$
we get that the reconstruction error fulfills
$$
\|x_\alpha^\delta - x^\dagger\| \to 0\quad\text{if}\quad \delta\to 0,\ \alpha\to 0,\ \text{and}\ \tfrac{\delta}{\sqrt{\alpha}}\to 0.
$$
One can get quantitative estimates for the reconstruction error by assuming so called source conditions, e.g. that $x^\dagger$ is in the range of $A^*$. With this assumption one can estimate the approximation error and arrive at estimated of the type
$$
\|x_\alpha^\delta\|\leq C_1\tfrac{\delta}{\sqrt{\alpha}} + C_2\sqrt{\alpha}
$$
which shows that one get "best reconstruction" of one chooses $\alpha$ in the order of $\delta$.
The constants $C_1$ and $C_2$ do not depend on the condition number of $A$. To be more precise: $C_1$ is fully independent of $A$ (for Tikhonov regularization it is $1/2$), but $C_2$ depends on $A$ implicitly. For Tikhonov regularization one has $C_2 = \|w\|/2$ where $w$ is such that $x^\dagger = A^*w$ - but the condition of $A$ is not so important - it's more the norm of $A$.
