A thinner béarnaise?
The thickness is coming from the ratio of oil to water. Like mayonnaise, bearnaise is an oil-in-water emulsion, and will get thinner the more water is in it. (Consider the relative thickness of moist sand and mud; the butter in the bearnaise is like the sand.) Adding a bit of hot water will work well to thin it, but do so a bit at a time, stirring well afterwards, as a little bit of water will go a long way.