Have a look at

Bayesian Results for the Inverse Gaussian Distribution with an Application

Asit K. Banerjee and G. K. Bhattacharyya, Technometrics

Vol. 21, No. 2 (May, 1979), pp. 247-251

http://www.jstor.org/stable/1268523?seq=1#page_scan_tab_contents

and

Bayesian Estimation of Inverse Gaussian Distribution

B. N. Pandey† & Pulastya Bandyopadhyay

available from http://arxiv.org/pdf/1210.4524.pdf