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There is a class of interpretations of quantum theory which take a so-called epistemic perspective of the wave-function. In the epistemic perspective, the wave function represents an agent's knowledge, belief or information regarding a system.

This has a lot of appeal partly because wave function collapse looks very similar to the way that probabilities are updated in bayesian statistics when new information comes to light. For example, if I measure the polarisation of a photon in the vertical component and get a "yes" results (V1=y), then I have complete confidence that future measurements of the vertical will again get a "yes" result (V2=y), whereas future measurements of the horizontal component will be 50-50 between a "yes" result (H2=y) and a "no" result (H2=n).

P(V2=y | V1=y) = 1

P(V2=n | V1=y) = 0

P(H2=y | V1=y) = 0.5

P(H2=n | V1=y) = 0.5

You can sum this up as "the wave function is a way for me to keep track of probabilities, rather than an objective fact about the system".

Of course this idea has its own set of challenges. For many people the question becomes "what does the wave function represent knowledge of"? Also, a few years ago there was a thought-experiment known as the PBR theorem, which rules out some epistemic interpretations: https://en.wikipedia.org/wiki/PBR_theorem

There is a good set of slides on the PBR theorem and its implications by Jon Barrett here (he was my external viva examiner): http://www.cs.ox.ac.uk/qisw2012/slides/barrett.pdf

One class of epistemic interpretations that persists are the so-called bayesian interpretations of quantum theory: https://en.wikipedia.org/wiki/Quantum_Bayesianism .

A "strong" version of quantum bayesianism is QBism, which asserts that quantum theory is no more than an effective user manual for organising subjective beliefs about the outcomes of future interactions with the world. Chris Fuchs and Ruediger Schack are two prominent theorists advocating QBism.