I don't know much about Cronbach alpha but my guess is that there's no mathematical reason that .7 is the cutoff. There are a lot of 'rules of thumb' when it comes to cutoff values and there's no reason for most of them except that at some point somebody decided it was a decent value to use.
Alpha of .05: We get this because Fisher one time said that an observation that is more extreme than what we'd see in one out of twenty cases should be considered significant.
VIF > 10 means multicollinearity (some people use other values): Totally empirically based (and the fact that variables giving an R^2 > .9 can give us a VIF > 10). You can do whatever you want with this but it suggests there might be some issues with correlation among the independent variables.
Sample size greater than 30 to use a t-test on nonnormal data: Slight application of the CLT but 30 is nowhere near infinity. Mainly based on observations that for data that is even somewhat skewed the sampling distribution of the mean is approximately normal.
So I highly doubt that .7 is truly a magic number. But even though I don't know much about Cronbach's alpha my guess is that it's a decent rule of thumb that has been probably studied empirically.
I've seen .6 and I think .5 for "exploratory" research. I don't have reference but will check.
If you want to try and keep your scale and alpha is borderline 1) you can run factor analysis and report unidimensionality etc, if found, 2) if not found, throw out non-loading items and re-run, and/or 3) try to explain the low alpha. If there is little variance in the data, alpha will be small.
Thanks for all the feed back. I want to elaborate a little on the problem.
I have subscales of a larger total scale that have small cronbach alpha. I am afraid that removing the subscale will grossly alter, or make it difficult for others too interpret the meaning of the total scale.
AFAIK the 0.7 criterion was first (or at least most famously) suggested in: Nunnally, J. C. (1978). Psychometric theory. New York: McGraw-Hill.
I don't know of any particular strong empirical basis for the number - it's really just the recommendation of someone famous in the field. Given that the interpretation of alpha is somewhat dubious anyway (see here) it's hard to justify slavish adherence to a particular rule of thumb. Whether or not you keep the subscale has to be a decision made in the context of information from other sources - e.g. factor analysis results, the theoretical basis for the subscale and its use in your study, evidence for validity, psychometric results from other studies using the scale, the length of the subscale (short scales will have lower alphas), etc etc.
It could be useful to know the length of the subscale(s) of concern, and the nature of the response scale (Likert with x response options, dichotomous, etc?)