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I'm trying to add a random perturbation term to my response variable (to estimate the impact of multicolinnearity in the beta estimates), but I am not sure on how to accomplish it when the response variable is transformed using a 1/4 power, in R.

Without any transformation in the response variable, according to Faraway's "Linear Models with R," the way to write it is: 

g <- lm(hipcenter+10*rnorm(38) ~ ., seatpos).

So if I have a 1/4 power transformation for the response variable, then the code should be: 

g <- lm((hipcenter^(1/4))+10*rnorm(38) ~ ., seatpos).

Is this right?

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Why don't you construct another variable in your data.frame that is transformed and use that one in your model? Something along the lines of seatpos$hipcenter_trans <- seatpos$hipcenter^(1/4) + 10 * rnorm(38). To answer your question, yes, you are powering your variable and adding some noise that has mean of ~0 and SD of 10. – Roman Luštrik Aug 31 '12 at 8:35
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Are you aware of the perturb package in R? It seems to do exactly what you want. – Peter Flom Aug 31 '12 at 10:36
I just wanted to thank Roman and Peter for their helpful advice. – Dan Sep 1 '12 at 19:30

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