How many bits of information can be stored in a qubit? On making a measurement on a qubit, one can catch it in either $|0\rangle$ or $| 1\rangle$. Seems that measurement on a qubit can not reveal more than a single bit. But does it mean the a qubit can't store more than 1 bit of information?
No more than one. It follows from Holevo's theorem that $n$ qubits cannot be used to store more than $n$ bits of information. See for example these notes for an explanation of Holevo's theorem (interestingly, the main question answered by these notes is exactly the title of this question).
Very roughly speaking, the reason for this is that if you try to encode $n$ bits of information in a system of dimension $d<2^n$ (so, for example, in less than $n$ qubits), then there is no reliable way to retrieve such information. This is because if $d<2^n$, then there are too few orthogonal states on which to store the information (if two states are not orthogonal, they cannot be deterministically distinguished).
Note that this is very different than saying that a single qubit requires a single bit to be described. In fact, in general, you need an infinite amount of bits to describe the state of a single qubit, because a single qubit is characterised by continuous numbers. This means that, in some sense, a single qubit can encode an arbitrarily large amount of information. The problem is that you can't deterministically decode such information afterwards, using a single copy of said qubit. You could use many copies of the same qubits to completely reconstruct its state, and therefore recover all of the information encoded in it, but this turns out to be a pretty inefficient information storage scheme.
What is really contained in a single qubit are 2 real numbers. This is the amplitude of the $|1\rangle$ state (the other one is given by the normalization), and the relative phase of the two states. When measured, we get out a single bit of information, as you correctly stated, but with which probability we get this specific bit is determined by these real numbers. Measuring equivilantly prepared states multiple times allows us to infer something about the 2 variables contained in the state.